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Question: If a tangent to ellipse \(\frac{x^{2}}{a^{2}}\)+ \(\frac{y^{2}}{b^{2}}\)= 1 makes an angle a with x-...

If a tangent to ellipse x2a2\frac{x^{2}}{a^{2}}+ y2b2\frac{y^{2}}{b^{2}}= 1 makes an angle a with x- axis, then square of length of intercept of tangent cut between axes is-

A

a2tan2a + b2 cot2a

B

a2 sec2a + b2 cosec2a

C

a2 cosec2a + b2 sec2a

D

None of these

Answer

a2 sec2a + b2 cosec2a

Explanation

Solution

Let tangent is xcosθa\frac{x\cos\theta}{a}+ysinθb\frac{y\sin\theta}{b}= 1

Slope is tan a = –ba\frac{b}{a}cot q … (1)

\ square of length of intercept is

= (a sec q)2 + (b cosec q)2

\ Length is L = a2 + b2 + a2tan2q + b2 cot2 q … (2)

Now use value of tan q from (1)