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:
3 x 1 + 4 x 2 = 5
Soln.
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)Determine the polar (r, q) equation for :
x 1 2 + x 2 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
= r 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
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