1)Find
the coordinates of the center of each for the following circles and the radius
r. Sketch each of the circles
a) x 2
+ y 2 - 2 y = 3
b) 2x 2
+ 2 y 2 + x + y = 0
c) x 2
+ y2 + 2x = 8
Solutions:
a) x 2
+ y 2 - 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
So:
r 2
= (0
2 + (-2) 2
- 4(-3_) / 4 (1) 2
r 2
= 16/ 4 = 4 so radius r = 2
Sketch of the resulting circle:
b) 2x 2
+ 2 y 2 + x + y = 0
Here: A = 2, C= 2, D =
1, E = 1, F = 0
Center is at:
(h, k) = (-D/ 2A),
(-E/ 2A) = (- 1/4, -1/4)
Radius:
r 2
= D
2 + E 2
- 4 (AF) / 4 A 2
So:
r 2
= (1
) 2 + (1)
2 - 4(0) / 4 (2) 2
r 2 = 2 - 0 = 2
So: r = Ö
2 (Sketch below)
c) x 2
+ y 2 + 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)
Radius:
r 2
= (2
2 - 4(-8)) / 4 A 2
So:
r 2
= 36/ 4 = 9
So; r = Ö 9 = 3
The circle is as shown below
2) For ellipse: We have: A = 9, B= 0, C = 4, D=
36, E = -8 and F = 4
Then we get: 9 x 2 + 4
y 2 + 36 x - 8 y + 4 = 0
With appropriate algebraic
manipulation we get:
(x + 2) 2 / 4 + (y
- 1 ) 2 / 9 = 1
Graphing yields:
3) Determine the equation to which:
x2 + xy + y2 = 1
reduces when the axes are rotated to eliminate the cross product term. Sketch the resulting curve.
reduces when the axes are rotated to eliminate the cross product term. Sketch the resulting curve.
Solution:
The given equation has: A = B = C = 1
Choose Θ according to: cot 2Θ = (A - C)/ B = (1 - 1) / 1 = 0/1 = 0
Then: 2Θ = 90 deg so: Θ = 45 deg
Then: x = (x' - y') / Ö 2
y = (x' + y' ) / Ö 2
And, after some algebra:
3(x') 2 + ( y') 2 = 2
Divide through by 2:
3(x') 2 / 2 + ( y') 2 / 2 = 1
This curve defines an ellipse which is shown below:
Choose Θ according to: cot 2Θ = (A - C)/ B = (1 - 1) / 1 = 0/1 = 0
Then: 2Θ = 90 deg so: Θ = 45 deg
Then: x = (x' - y') / Ö 2
y = (x' + y' ) / Ö 2
And, after some algebra:
3(x') 2 + ( y') 2 = 2
Divide through by 2:
3(x') 2 / 2 + ( y') 2 / 2 = 1
This curve defines an ellipse which is shown below:
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