## Wednesday, May 4, 2022

### Solutions To Differential Geometry Problems (Part 3)

1) Sketch the graphs of;

x 2  = - Ö ( 8 2   -   x 1 22)

And:

x 2  = - Ö ( 16 2   -   x 1 22)

On the same Cartesian axes.

Soln.

2)(a) Write the polar form of the equation of the line:

x 1   +   4 x 2   =  5

Soln.

x 1 = r cos q

x 2   =  r sin q

then:  x 1   +   4 x 2   =  3 r cos q + 4 r sin q

The polar form of the equation is:

r [3 cos q + 4 sin q]  =  5

b)Determine the polar (r,  q) equation for :

x 1 2  + x 2   -  2ax 2  =  0,    a  ≠   0

And sketch the resulting curve.

Soln.

x 1 2    +    x 2 2  =   ( r cos q ) 2  +  ( r sin q ) 2

=   2 (cos 2 q   +   sin 2 q

=> (eqn. of circle)   r    =   2 a x 2  =   2a  sin q

3)(a)  Let r and  q  be polar coordinates in the x 1 x 2 - plane. Give the representation of the following curve in Cartesian coordinates:

r =   a q

And sketch it.

Soln.

(x 2 -   x 1)  tan  Ö ( x 1 2  +   x 2 22) /  a  =    0

This is an Archimedian spiral:

b)  A student's analysis of the curve (cardioid):

r =   6 (1 - cos q),    is shown below:

Using differential calculus show how an expression for the angle y  can be obtained in terms of the angle   q.

Soln.

r =   6 (1 - cos q)

dr =   6 sin q  dq

tan y   =  r dq /  dr

tan y   =  6 (1 - cos q) d q /  6 sin q  dq

=   tan q/ 2