Question

Let $Q(acos\theta ,bsin\theta )$ be a point on the auxiliary circle. Then the corresponding point with respect to Q on the ellipse when a line drawn perpendicular to major axis AA' will be.

• A. $(acos\theta ,asin\theta )$
• B. $(acos\theta ,bsin\theta )$
• C. $(bcos\theta ,asin\theta )$
• D. $(bcos\theta ,bsin\theta )$

Answer 1

The correct option is B $(acos\theta ,bsin\theta )$

A circle described on major axis of an ellipse as diameter is called auxiliary circle. For every $Q(acos\theta ,bsin\theta )$ on the circle if we drop a perpendicular to the major axis it will cut the ellipse at point P. These points P and Q are called corresponding points.
Now let's find P.
We know, $Q(acos\theta ,bsin\theta )$
X - Coordinate of P will be same as of Q i.e.,$acos\theta$
We also know that P is a point on the ellipse,
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[putx=acos\theta ]\Rightarrow \frac{a^{2}co{s}^2\theta }{a^2}+\frac{y^2}{b^2}=1\Rightarrow y^2=b^2(1-co{s}^2\theta )=b^{2}si{n}^2\theta \therefore y=bsin\theta \therefore \text{Coordinates of P will be}(acos\theta ,bsin\theta )$