15:I[8216,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"BackButton"] 16:I[1553,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"QuestionViewer"] 17:T58c,

Suppose OB makes an angle $$\theta$$ with the positive direction of the X-axis, then OA makes an angle of $$\theta+\frac{\pi}{2}$$ with the positive direction of the X-axis.

WB JEE 2022 Mathematics - Ellipse Question 16 English Explanation

Let $$OA=r_1$$ and $$OB=r_2$$ coordinates of A and B are $$(-r_1\sin\theta,r_1\cos\theta)$$ and $$(r_2\cos\theta,r_2\sin\theta)$$ respectively.

As B lies on the ellipse

$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

$$ \Rightarrow {{r_2^2{{\cos }^2}\theta } \over {{a^2}}} + {{r_2^2{{\sin }^2}\theta } \over {{b^2}}} = 1$$

$$ \Rightarrow {1 \over {O{B^2}}} = {1 \over {r_2^2}} = {{{{\cos }^2}\theta } \over {{a^2}}} + {{{{\sin }^2}\theta } \over {{b^2}}}$$ ..... (i)

Similarly, as A lies on the ellipse

$${1 \over {O{A^2}}} = {1 \over {r_1^2}} = {{{{\sin }^2}\theta } \over {{a^2}}} + {{{{\cos }^2}\theta } \over {{b^2}}}$$ ..... (ii)

From Eqs. (i) and (ii), we get

$${1 \over {O{A^2}}} + {1 \over {O{B^2}}} = {1 \over {{a^2}}} + {1 \over {{b^2}}}$$

Ellipse

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