Sunday, January 22, 2017

Solutions to Analytical Geometry Problems (2)

Problems: 1) In the last example problem worked above, we saw that the ellipse equation of form:

(x + 2) 2 / 4 +    (y - 1 ) 2   / 9 = 1

Could be simplified to: x' 2 / 4 +    y ' 2   / 9 = 1

Using:   x' = (x + 2) and y' = ( y - 1)

Show how the offset affects the location of the simplified ellipse form by graphing both ellipses on the same axes. Thus, satisfy yourself that the graph shown is justified once the x, y- coordinate offsets are taken into account.


The two ellipses graphed on the same axes are shown below:

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where the "primed" form of the graph is the one that is centered at (0,0).

2) An ellipse has center coordinates at C(-3,0) and focus at F(-3, -2) with a = 4. Find the eccentricity e and sketch the graph.

We have for coordinate-y difference:   (0 - (-2)) = 2 so c = 2

The eccentricity e = c / a    = 2/ 4   = 0.5

b =   Ö (a 2  -   c 2)    =   Ö (16  -   4 )    =  Ö 12

The equation is then (offset 3 units in x coordinate):

(x + 3) / a 2 +    y2  / b 2 = 1


(x + 3)2 /16 +    y 2  /  12  = 1  (see below)
No automatic alt text available.

3) An ellipse has center coordinates at C(2, 2) and focus at F(-1, 2) with a = Ö 10. Find the eccentricity e and sketch the graph.

e =  3/  Ö 10

Graph is:

No automatic alt text available.

4) Sketch the following ellipses from their equations below, and give the semi-major axes, semi-minor axis and eccentricity e.

a) 4 x2  +   9  y2   = 144

Solution for this graph:

No automatic alt text available.

From which we see: a = 6 units, and b =  4 units

Then: c  =   Ö (a 2  -   b 2)   =      Ö  (36   -  16)   =   Ö  20  

eccentricity e =   c/  a   =  Ö  20   /  6  =  0.745

b) 4 x2  +    y2   =    1

Solution for this graph:

No automatic alt text available.

From which we see: a = 1 units, and b =  1/2  unit

Then: c  =   Ö (a 2  -   b 2)   =      Ö  (1   -  1/4)   =   Ö  (3/4)   =  Ö 3/  2

eccentricity e =   c/  a   =  Ö 3/  2   /  1  =  0.866

5) Look up the definition of the "directrix" of an ellipse. Then find the eccentricity and directrices of the ellipse:

x2 /7 +    y2   / 16 = 1

And sketch the graph showing the locations of the foci.


Each ellipse, with two foci must have two associated directrices. One is shown below:
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Each directrix being perpendicular to the line joining the two foci then the eccentricity can be interpreted as the position of the focus as a fraction of the semimajor axis. 

The graph is shown below:

No automatic alt text available.

Let   a  =  Ö 7  

b =  4

=   Ö (b 2  -   a 2)   =      Ö  (16   -  7)   =   Ö  (9)   =   3

eccentricity e =  3/4    = 0.75

Directrices at:  y = 16/ 3 

Women's March Trumps Trump Inauguration - And Trumpies Go Ape Shit

Related imageImage may contain: 1 person, crowd, sky and outdoor
Don't you dare believe your lyin' eyes! You believe what my Twitter feed tells you! Or Sean Spicer, my chief , ass-kissing lackey!

The scenes from the Women's March yesterday in  Denver  as well as Washington, D.C. were captivating to behold. So many women taking to the streets to make their voices heard from every walk of life. The Washington  march wasn't the only one held in the US, however. From Atlanta to Phoenix, from Boston to Sacramento, “sister marches” staged a show of defiance by ordinary citizens determined to rebuke Trump’s values.  In addition, one beheld similar scenes from cities around the world, from London, to Prague, to Berlin. Organizers in London said 80,000 had taken part in an event there. In Prague, hundreds gathered in Wenceslas Square, waving portraits of Trump and Russian president Vladimir Putin.

These images and others warmed the cockles of our hearts, especially Janice's. She had been on a downer since Friday, unable to watch anything pertaining to the Liar-in- Chief's installation. This, even as we were having breakfast at a hotel in Aurora, getting ready to leave after my 3D biopsy.

So yesterday's magnificent scenes of the estimated 500,000 women on the march had the equivalent effect of a nice psychological balm - scenes which we will put on a DVD to remind us how many oppose this clown in office. Did I say clown? Uh, yes, indeed I did. Especially after this walking talking turd had ambled to CIA HQ in Langley, VA and actually stood at the site of their Memorial Wall - honoring 117 of the fallen - over decades.

He delivered a 16 minute spiel, about the same as his Inaugural PR piece, and used nine full minutes to whine about the media instigating the "feud" with the CIA (no, we have his Twitter feed records to show this fucker did it all by his lonesome) and then also whined about the media "lying" over the women's march crowds compared to those for his inauguration.

My thoughts on seeing a replay of his remarks on CNN were actually embodied in a question:

"Has this deplorable excuse for a human  no decency, no shame?  Has he no recognition of where he is? Has he not even the perceptual fine tuning to pay homage to the shrine whereupon he stands? Not even for five minutes? Are his petulant complaints worth more than those 117 lives?"

And today I stand by those words even more than yesterday. As one media commentator put it, "Is this going to be how it is? Complaining about every little perceived slight each day? What about running the country?"

Well, I am afraid this POS will leave that to his basket of despicables being streamlined to take over a number of agencies and run them into the ground.

Anyway, it evidently wasn't enough for Trump to spew his lies, he had to get his ass licking toady press imp Sean Spicer to pile on too (with uh, "alternative facts" to use the euphemism of Bimbo Troll Kellyanne Conway) . It seems the stark images of the Friday fiasco being outdrawn by the women's march as well as Obama's previous inaugurations caused the Trumpies' heads to explode.

In case you missed it, Spicer actually accused journalists of reporting inaccurate crowd numbers and using misrepresentative photographs to "minimize enormous support" for Trump. Which is a pile of horse merde.  In fact, actual dated images of the National Mall on Friday contradicted Spicer's assertions especially compared to Obama's two previous inaugurations..  Indeed, a time lapse video produced by PBS showed the National Mall was never full at any time on Friday.  Meanwhile, photographs show the National Mall packed with crowds just before Obama’s inauguration in 2009 – but much more sparsely covered with people just before Trump’s ceremony.

Spicer also claimed there's never been the white ground coverings on the National Mall before. But in fact photos from 2013 clearly shows National Park Service workers unloading them  for the Obama inauguration.  Spicer also lied by saying that hundreds of magnetometers prevented many more thousands from getting through to the event. But the Secret Service in fact noted there were no such devices for screening at all.

Spicer also tossed out lies to do with Metro Ridership numbers, inflating the numbers for Friday. The actual numbers were:

570,557  for Jan. 20, 2017

782,000 for Jan. 20, 2013

1.1 million for Jan. 20, 2009

Thus, Spicer's claim that the Trump numbers were the "largest ever" was another bald-faced lie, similar to Trump's that he actually won the popular vote.

As one media commentator put it:  "I just can't understand why he (Trump) decides to get up in the morning and go on these rants".

But we do KNOW why! Because we have a guy as leader of the free world who is chronologically 70 years of age, but emotionally at the level of a dyspeptic three year old who just soiled his pampers and wants them changed.

CNN's Jim Sciutto perhaps put the most somber perspective on the issues last night. He said that ultimately these numbers - of whose crowd was biggest' will not matter, and indeed fall into a memory hole. But what will matter are those like U.S. soldiers killed in the Middle East, North Korean missiles fired, or terror cells in the U.S. Will the Trump bunch lie (offer "alternative facts")  about all these too, if they can so easily do so about meaningless crowd numbers for events?

As Sciutto put it:

"When those facts and numbers matter, that have a real effect on our lives and national security are you going to trust the words coming from the president and the podium?"

Historian Douglas Brinkley's take is also germane,  referencing how LBJ and Nixon fudged the numbers on Vietnam war casualties:

"So Donald Trump has a sacred obligation to tell the truth."

But will he? I doubt it, because this is part and parcel of Trump's premoral character, e.g.

As I noted in my Jan. 13  post:

"While diehard Trumpsters may love this sort of reactionary behavior, it doesn't comport with strength especially in a President. What it shows is that any twerp, misfit, or non-political person (without political power) can get under the big man's skin and make him freak out. In other words, it shows he is hostage to any insult that comes down the pike and from any direction."

In other words, Trump's compulsive lies are driven by his thin skin syndrome. Because he has no tolerance for criticism and the facts to support him are few or non-existent, he must conjure up pseudo-facts, fake news and lies. The greater the truth and facts cast in his path, the greater his need to summon lies, fake news to try to try to counter them. In the end, it's a losing wicket which will ultimately undermine Trump's time in office because his credibility will be destroyed. Indeed, it already has been his stream of stupid, reckless tweets.

Without credibility no president has a leg to stand on, and certainly no moral character to either unite or summon the nation to common action.

Buckle your seat belts as it will be an intriguing 4 years indeed. Trump and his minions need to be on notice that they will be called on their lies and bullshit every time they spew it.

Saturday, January 21, 2017

Sollutions to Analytic Geometry Problems (1)

The problems again - with solutions to follow:

1)  Complete the square for the general analytic equation:

Ax 2    +  Ay2   + Dx  + Ey  + F   = 0

To show how the radius r is derived, as well as the coordinates for the center of a circle not necessarily at (0,0)


Write:  (x2   + Dx / A) +  (y2   + Ey/ A) =    - F/A

[x2   + Dx / A  +   (D/ 2A)2  ] +   [y2   + Ey / A  +   (E/ 2A)2 ]

= - F/A + (D 2   +    E 2  /  4 A 2    


(x  +   D/ 2A)2   +  (y  +   E/ 2A)2       =   D 2  +   E 2      - 4AF) /  4 A 2 

Then:   r 2     = (D 2   +    E 2      - 4AF) /  4 A 2    

Looking at the two factors we see the center coordinates must be:

h =  (-D/ 2A)   and k =  (-E/ 2A)

2) Check the equation:  x 2    +    y2  =  4

to see that the general form also applies here.


Compare to general form:

Ax 2    +  Ay2   + Dx  + Ey  + F   = 0   (A not equal 0)

Then: A = 1 , D = 0, E = 0, F  =   -4

The center is at:

h = (-D/ 2A)   = 0   and k =  (- E/ 2A) = 0  So:   C(0, 0)

Since   4  =  r2  then r  is 2, then we see radius is 2.

3) Find the coordinates of the center of each of the following circles and the radius r, also sketch the circle:

a)  x 2    +    y2  - 2 y   =  3

We have: A = 1, D = 0,  E = -2  and F = -3

Then the center is at: 

(h, k) =    (-D/ 2A),   (-E/ 2A)  =   (0,  1)

r 2     = (D 2   +    E 2      - 4AF) /  4 A 2    


r 2     = (0 2   +    (-2)  2      - 4(-3_) /  4 (1) 2    

r 2     =   16/ 4   = 4  so radius  r  = 2

No automatic alt text available.

b) 2x 2    +  2  y2   + x  + y = 0

Here: A = 2,   D = 1,  E = 1,  F = 0

Center is at:

(h, k) =   (-D/ 2A),   (-E/ 2A)  =   (- 1/4,  -1/4)


r 2      = (D 2   +    E 2      - 4AF) /  4 A 2    


r 2     = (1 2   +    (1)  2      - 4(0) /  4 (1) 2    

=   2/ 4 =  1/2 

So:  r =  1/ Ö 2  

No automatic alt text available.

c) x 2    +    y2  + 2x  = 8

We have: A = 1,  D= 2, E = 0, F = -8

Center is at:

(h, k) =   (-D/ 2A),   (-E/ 2A)  =   (- 2/ 2,  0 ) = (-1, 0)


r 2      = (2 2     - 4(-8)) /  4 A 2    


r 2     =    36/ 4   =  9

So; r =  Ö 9   =    3

The circle is the same as shown below for (4)

4) Obtain the equation for the circle shown below:
No automatic alt text available.
By inspection, center is at (h, k) =   (-1, 0)

Also, radius r = 3 units

We now write the most general form using coordinates (h, k) for the center:

(x - h)2    + (y - k ) 2   =  r 2 


(x - (-1))2    + (y - 0 ) 2   =  3 2 

The equation for the circle is then:

(x + 1)2    + y 2   =   9


x 2    +  2x  + 1  +    y2   = 9 


x 2    +    y2  +  2x   =  8

Trump's "American Carnage" - Methinks He's Got The Wrong End oF the Stick

Graph showing how national debt would blow up if Trump's tax cuts are enacted (Source: Committee for a Responsible Federal Budget)

As Trump crouched this morning in an Episcopalian church  pew trying not to let his mug be seen at a "national prayer service"   nearly a million women - many in "pussy hats" - are ripping him to shreds as they march in D.C. - and around the world. Trump yesterday used a 16 minute speech to howl that "American carnage" is about to halt under his watch, careful not to tell his groupies he will be the major source of it.  So,  why not rip a page from Hitler's playbook and use others as scapegoats for what is about to befall the millions who blindly voted for him? Works every time, at least for a while.

Don't take my word, just think of the coming consequences as a result of his assorted appointments of "despicables" and the announced policy changes  To fix ideas, consider his tax proposals alone. He has demanded an "across the board" tax cut. It starts by increasing the current standard deduction of $12,600 (for joint filers) now, to $30,000.  These deductions, by the way, are what you use when you don't itemize.   At the same time, Trump would dump the $ 4, 050 write off claim for each member of a household, e.g. the "personal exemption". He also wants to simplify the tax code by changing from the current seven brackets - ranging from 10 percent to 39.6 % to just three brackets.

In addition, if he gets the help needed from congress, he will scrap in turn: 1) the AMT or alternative minimum tax, 2) the estate tax and 3) the 3.8% surtax on investment income - as part of abolishing Obamacare.  As regards the corporate tax rate he would lower it from the current 35 % (actually few businesses pay that rate, most are effectively 5%) and replace it with a 15 percent rate.  He has also floated the idea of applying this not only to big corporations but small business owners and the self-employed, including dentists, law partners, and Trump himself.

While all this sounds terrific in theory it carries a steep price tag, which you can get an idea of by studying the graphic at top. That is, the federal debt as a percentage of GDP would explode through the roof - exceeding the size of the entire U.S. economy within ten years. The debt incepted - in order to be even relatively controlled- would necessitate draconian cuts in every federal "entitlement" program including: Social Security, Medicare, Medicaid, Tricare, and all VA programs, departments. With such astounding debt there will be no "winners".

You want carnage? That's carnage - and it ain't coming from the global trade order but from Trump's own administration!

Then there is the soon to come painful loss of health care by 18 to 20 million Americans, nearly two thirds of whom voted Trump.  Many of these people still can't believe it, but mainly because they confused "Obamacare" which they pillory and mock - with the ACA - which they somehow think is different. Hence, not realizing the ACA ("Affordable Care Act") and Obamacare are one and the same.

At the heart of the Trump signing repeal is to get rid of the mandate, through which those who have more health issues can be covered by those with fewer. Once that mandate is removed then the protections of the ACA fall through. First among them will be that pre-existing conditions will now be reckoned into charges or whether people get insurance at all. Oh yeah, you may get "access" - but have to pay $200k a year to get it!  Then  there is the other negative that young people currently allowed on their parents' plans to age 26 will no longer be covered.  They'd have to pay for their own plan. And what is on offer? Well, according to Rand Paul: "those taken off the ACA can get a high deductible plan or use tax credits combined with a health savings account".

If you believe that doesn't spell c-a-r-n-a-g-e, think again!

Now, if you want REAL carnage, watch spellbound as Trump's minions in the EPA (like Scott Pruitt)  enable wholesale fracking around the country, including in national parks- and try to extract every last erg of oil from the ground. This will come on the latest NOAA and NASA reports that 2016 was the third hottest year in a row and making the planet the warmest it's been in 125,000 years.

Carnage? Oh yeppers, plenty! From power grids going down from overuse during extended heat waves, to ocean bacteria and plankton decimated, to the spread of tropical diseases like dengue fever and worm infestations, including of the brain. It is expected that before the first year of no seasons (est. 2040) more than 3 billion humans will bear worms in their brains. Alas, these will likely infect - with their larvae- as many Trump voters as others.

Trump insisted in his spiel he could repeal the global trade order by simple decree or his imperious say-so, but if any believe that they are fully delusional. It is not going to happen. Trump, rather than find scapegoats, would have done better to level with his followers that they all need to make common cause with the rest of us (non-Trump voters)  - to halt the climate nightmare unfolding and threatening everyone.

Sadly, Trump's ego and Id were too inflated to do that.  Hence, when Gloria Steinem addressed the D.C. crowd this morning she commented on Trump’s inauguration speech yesterday.:

Everything that happened before him was a disaster and everything that he would do would be fantastic, the best ever, miracles and all the superlatives."

Leaving out, of course, that he - Trump - will be the one to visit the carnage -  which will likely fall heaviest on his own voters.

Miracles? Trump's people, his voting blockade, will need them in spades within months.

Bank on it!

Thursday, January 19, 2017

3D Staging Prostate Biopsy Unlikely To Come Into General Use - My Account

Image of a 3D reconstruction- with grids superimposed-- after 3D Staging Biopsy. Note the prostate (in red), urethra (in green) and the exact location of areas of cancer (in mud yellow). 

The advantages of the 3D staging biopsy for prostate cancer which I just had two days ago, are well known and explicated wonderfully on Dr. E. David Crawford's University of Colorado site. We are informed, for example, that with this 3D rendition many more lesions, prostate cancer tumors can now be identified because the technique is much more thorough. As Dr. Crawford has noted, from 10-30 percent of prostate cancers are missed even in the MRI fusion biopsy.

In other words, given the much higher accuracy the 3D Prostate biopsy is now the "gold standard" for PCa diagnosis. But that standard doesn't mean many more men will choose it.  We need to look at why this is so.

The standard prostate biopsy is more or less an "in and out" procedure, done rectally with the accompaniment of an ultrasound to provide location data for the prostate sampling. This may be from 10 to 20 cores. Thus, a needle sample device is inserted into the rectum and each time extracts tissue from the gland after puncturing the rectal wall. Primary risk is for sepsis, which is why antibiotics like ciproflaxin must be administered in advance.

The procedure is usually done in the urologist's office without even a sedative and takes from 20 minutes to 45 minutes, depending on the number of sample cores taken. The chief side effects include: swelling and pain in prostate, possible problem urinating, and blood in urine, stools and semen. In the former two body fluids it can remain 1-2 days and in the last up to 3 months.

The 3D staging biopsy by contrast is an entirely different animal. While described as an "outpatient" procedure, you are actually administered general anesthesia because the technique is much more invasive and entails up to 100 needle sampling insertions through the perineum to extract prostate tissue. The locations are again confirmed via trans rectal ultrasound for later mapping of all the samples to a 3D grid. The samplings are taken approximately 5mm apart. In my case a total of 60 were taken or roughly a factor two the prostate mass (31 g).

Of course, before getting it done you are undergoing a full surgery prep, including being asked if you're an organ donor, and to leave an advanced medical directive (or living will) in case the surgery or anesthesia goes awry - or you hemorrhage. Also Janice had to be there to provide durable power of attorney and medical power of attorney in the event I was left unable to render decisions, directions for my own care.

In the pre-op setting you also discuss with the anesthesiologist any allergies, and note if you are sensitive to meds (as I am.) By the time you are being wheeled into the OR you are already half out of it from the mix of sedatives, painkillers including fentanyl.

Once in the OR you must move yourself from the gurney to the operating table where the sampling is done. I don't believe I'd completed the transition more than a minute before I was knocked out.   (Just before I was asked by a urology resident if I wanted a transfusion in case of hemorrhage, and I gave my blood type).

I awakened in the PACU or Post-Anesthesia Care Unit, to the soft voice of a beautiful nursing grad student, "Rachel", who asked the usual questions: Where are you? What is your name? What is your date of birth? What year is it? Etc. All this is done to ensure you are all there.

She then questioned me about the pain, where it hurt and the nature. I told her the worst aspect was a burning sensation all through the urethra, from the catheter. (I ended up opting for a Foley catheter, as opposed to supra-pubic on the advice of a urology resident, "Tim".).

She immediately brought me three meds to take with water, one of which was specifically to reduce the burning sensation that had become almost unbearable.

After giving me the meds, she brought me a few light Jello snacks. This time (unlike my gall bladder removal in May) there was no nausea at all so the anesthesiologist hit just the right mix.

Janice by now had also come up, after tracking my various locations using a "patient tracker" - coded by color to follow my progress. Once she arrived, Rachel and another student RN ("Callie")  showed her how to change the catheter bags, and also go from the usual (day time) leg bag to the much larger night bag.

Let me say right off the bat that having a Foley catheter in place is no ball of fun, and ranks maybe up there with a root canal - or two.  I had to wear the damned thing for two days before Dr. Crawford's RN gave the ok to take it out.   She emphasized it was foolhardy to do so too early because then "you could be in a real world of hurt". That refers to all the ancillary tissues swelling up preventing urination, which earns you a trip to the ER. 

So now, with it out I am still recovering and will get the results of the biopsy in 5 days or so.

Having had the normal biopsies and this one I can agree right now with what a paper dealing with it noted, that in general most men would not opt for it given the greater "medical burden" - that entails more invasive procedure - as I described - as well as more side effects, bleeding from rectum, penis, etc. and pain as well as having to wear a urinary catheter.

Most men, despite the touted accuracy of results (which isn't in dispute),  will simply not opt for this advanced biopsy unless they are looking for a specific treatment option (e.g. focal cryo-ablation) for which it is required.

Anyway, recovering now, and thankfully with no catheter. I am just waiting for further word on whether I am a candidate for the focal cryo. I will have to meet with Dr. Crawford again in 2 weeks.

See also:

Wednesday, January 18, 2017

"Democracy Is At Risk"- John Dean's Take On Trump Is Terrifying - Except for the Trumpists

No automatic alt text available.
"Remember, Donald, we have the photos of you with our beautiful Russian whores!"

Former Nixon legal advisor John Dean definitely got Chris Hayes' attention on 'All In' last night in terms of the threat Trump poses to democratic institutions in the country. Dean's appearance was instructive, given his testimony helped bring down one of the worst state presidential criminals in history: Richard Milhous Nixon.

For those not aware at the time, it is incumbent to get hold of the Nixon Whitehouse tapes, which ultimately did the bastard in. It was those tapes, replayed in parts during the Watergate hearings in 1973 that incriminated Tricky Dick. Hostile to those arrayed against him, Nixon created his own "enemies list" and then weaponized branches of the government -including the FBI, IRS - to go after those enemies (such as Daniel Ellsberg).   The Watergate break-in itself was Tricky Dick's final undoing.

But at least at that time, enough of Nixon's own party had the integrity to "read him the Riot Act". Thus, four GOP members of congress went to tell Nixon he had no choice other than to resign, and resign he did. In addition, the robust press at the time - namely The Washington Post with its Carl Bernstein and Bob Woodward- did true investigative reporting to get at the bottom of this conspiracy.

Dean pointedly noted the same checking forces are now much weaker, and not likely to put a halt to Trump's overreach - where he will basically wield more power than any other previous holder of the office. For example, today the members of a party (GOP) prepared to condemn the actions of Trump can be reduced to one finger of one hand. (Rand Paul refused to vote for repeal of the ACA)

In addition, the media today is only a weak sister of what it was in the Watergate era. As papers have been sold off or merged, and support of true investigative journalism pulled back, PR has now come to dominate. The alert reader need only scan through the pages of his local newspaper to behold all the sources that are either PR outfits, or think tanks - with the mandate to bend the news and readers' perceptions.

The absence of these cornerstones for scrutiny is important given the Trump program is orders of magnitude worse than anything Nixon conceived. For starters, Trump's enemies list will probably contain most of the nation's Left bloggers, as well as those organizations (like the SPLC) that inveigh against racism - especially calling fellow humans "apes" as a replacement euphemism for the N-word.

Also, while the damage inflicted by Nixon was mainly from his inner circle, the devastation wrought by Trump's derelicts will be vast given it will include rabid enclaves laden with Alt-Right Nazis, KKK, Aryans and the like. Some, under the "Bikers for Trump" helm, have vowed to crack open heads and harass anti-Trump protesters present on Inauguration Day.  I suggest that any protesters encountering them have their own cans of Mace and solid helmets on for protection.

In similar fashion, Dean pointed out the only way a fascist Trump agenda might be stopped is from the bottom up - in other word, enraged citizens venting their furor, especially to state Senators, congress critters. As Dean observed, once their own re-elections are threatened they will see no point in blindly backing Trump.

But at the same time, Dean emphasized that nearly 63 million of our fellow citizens will never lift a finger or  say anything because they actually applaud what Trump is doing, i.e. to tear down the establishment. Or so they've been told. What I want to see is what they do when Trump and his acolytes take a sledge hammer to their health care, veterans' benefits and even Social Security.

Against this background, it was heart warming to learn that in the coming days more than 1 million people in 32 nations and 51 states plan to protest against the illegitimate Trump ascension.  The problem is that these protests cannot be a one off. No they instead must be used as a basis for a massive networking of a Trump resistance.  That resistance will be in charge of national strikes, boycotts, as well as new, energized voter registration efforts.

The message of a new resistance -protest movement will be "Take Down Trump By 2018!" Of course, that most likely will be via the 25th Amendment route which will be invoked as Trump shows clearly he's unfit to lead, likely because of a serious mental disorder (intermittent explosive disorder and schizoid behavior)

That will be important, as any Trump reign lasting overly long will likely mean the end of this country as we know it. Ever seen the film 'Idiocracy'? You should!

Monday, January 16, 2017

A Look At Analytic Geometry (3): Parabolas

From parabolic telescope mirrors to receivers, the parabola plays a major role, and so it is useful to study the properties of this particular figure. Probably the classic parabola of form:

y  =   x2   

is a good place to start. A graph of this equation is shown below:

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Note the vertex is on the origin or (0,0), but as with the circle and ellipse this need not be so, and one can encounter parabolas that are displaced, e.g. away from the x-axis, as well as oriented in a different way. For example, consider:  x2  =  -  y   

And the graph shown below:
No automatic alt text available.

One can also obtain parabolas with vertices oriented to the left or right of the y-axis. It is left to readers to try to find equations that yield these.

The general form of the parabola is often given as:   x2  =  -  4py 

if the parabola opens downward as shown above. If the reverse is true it is given as: x2  =  4py  . In each case, the focus of the parabola is on the axis of symmetry (e.g. follow the y-axis through the vertex downwards - or upwards in the earlier case) and p units from the vertex.

Let's consider the parabola shown  for   x2  =  -  y    then:

- 4p =   1  and p = - 1/4

That is, -  1/4 unit from the vertex to get to the focus F. What would we have for the previous parabola? Well, you should be able to work out: p = 1/4

Now, the directrix of the parabola will be the line y = p or y =  -1/4 for the downward oriented parabola. It is a line parallel to the x-axis and by that amount of displacement.  It should therefore be no surprise that the tangent to a parabola at its vertex is parallel to the directrix.  Consider a parabola of form:

x2  =   4py 

then the slope of the tangent at any point is dy/dx = x/ 2p. which is 0 at the origin.  But the second derivative:  d2 y /  dx2   = 1/2p which is positive.(So the curve is concave upward).

Of course, not all forms for the parabola are as straightforward as shown. Consider the equation:

2 x2  +   5y   -3x + 4 = 0

We want to try to get this into the more general form for the parabola:

(x -   h) 2  =   - 4p (y -  k)

We can begin by dividing both sides of the initial equation by 2:

 x2  +   5y /2   - 3x/2  + 2   = 0

Which can also be written as:

x2     - 3x/2  =    - 5y /2   -  2

We can then complete the square, i.e. adding (- 3/4)2   to both sides:

x2     - 3x/2   + 9/16  =    - 5y /2   -  2   + 9/16

Or, write as:

(x  - 3/4)2     =   -5y/ 2  - 23/16

We can factor out 5/2 from the right hand side, i.e.:

(x  - 3/4)2     =   -5/ 2  (y   + 23/ 40) 

Now, compare this to the general form we provided:

(x -   h) 2  =  - 4p (y -  k)

From this we see that:  h = 3/4,  k =  -23/40  and 4p = 5/2  so: p = 5/8

Therefore, the following hold true and can be confirmed by sketching the graph (see problem 1):

a) The vertex is situated at V(3/4, - 23/40)

b) The axis of symmetry as always is x = h or x = 3/4 in this case.

c) The focus is p units below the vertex at F(3/4, - 6/5)


1) Graph the parabola (x  - 3/4)2    =   -5/ 2  (y   + 23/ 40)   and defined by the properties given above.

2)The parabola shown below:
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Has the general form  (y -   k) 2  =  - 4p (x  -  h)

Use this information to: a) find the actual equation of the curve, b) the coordinates of the focus, c) the equation for the directrix.

3) Find the focus and the directrix for the parabola: x2  =   8y

And graph it, clearly labeling the axis of symmetry.

Take the 2nd derivative before graphing and establish whether the parabola is oriented concave downward or upward.

4) Do the same for: y = x2  +   4x

5) Given  V(-3, 1) and F (0, 1) find the equation of the associated parabola and also for its directrix. Sketch the graph showing the focus, vertex and directrix.

6)  Find the tangent to the parabola y  =   x2     using the fact that the equation of the line tangent to a curve y = f(x) is given by:

y - y1 = f'(x1) (x - x1)

where (x1, y1) = P1 is the point of the curve for which the tangent is constructed and y1  =   (x1)2     .  Show the graph of the given parabola and the tangent line with the point P1 (x1, y1) identified.