009C Sample Final 3, Problem 7

A curve is given in polar coordinates by

$r=1+\cos ^{2}(2\theta )$

a) Show that the point with Cartesian coordinates $(x,y)={\bigg (}{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}{\bigg )}$ belongs to the curve.

b) Sketch the curve.

c) In Cartesian coordinates, find the equation of the tangent line at ${\bigg (}{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}{\bigg )}.$

Foundations:

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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