Solutions To Differential Geometry Problems (2)

 Problems:

1) Sketch the graphs of;

x ₂  = - Ö( 8 ²   -   x ₁ ²)

And:

x ₂  = - Ö( 16 ²   -   x ₁ ²)

On the same Cartesian axes.

Solution:

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

3 x ₁   +   4 x ₂   =  5

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

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

And sketch the resulting curve.

Solutions:

(a) x ₁ = r cos q

x ₂   =  r sin q

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

=   r ² (cos ² q   +   sin ² q) 

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