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CBSE Class 11-science Answered

find the locus of the middle point chord of an ellipse X^2/A^2 + Y^2/B^2 =1 which substends a right angle at their centre
Asked by harshthakkarht2001 | 07 Jan, 2019, 05:33: PM
answered-by-expert Expert Answer
Let a chord PQ subtends an angle 90° at origin O in an ellipse begin mathsize 12px style x squared over a squared plus y squared over b squared space equals space 1 end style
Let the parametric coordinates of P be (a cosθ, b sinθ).

Slope of OP = begin mathsize 12px style fraction numerator b space sin theta over denominator a space cos theta end fraction end style  ;  hence slope of OQ = begin mathsize 12px style negative fraction numerator a space cos theta over denominator b space sin theta end fraction end style
Since OQ passes through origin, coordinate of Q is given by ( b sinθ, -a cosθ)
 
if R is mid point, then coordinate of R (x, y)  is begin mathsize 12px style open parentheses fraction numerator a space cos theta space plus space b space sin theta over denominator 2 end fraction space comma space fraction numerator b space sin theta space minus space a space cos theta over denominator 2 end fraction close parentheses end style
begin mathsize 12px style H e n c e space fraction numerator x plus y over denominator b end fraction space equals space sin theta space space space a n d space fraction numerator x minus y over denominator a end fraction space equals space cos theta H e n c e space open parentheses x minus y close parentheses squared over a squared space plus space open parentheses x plus y close parentheses squared over b squared space equals space 1 space space space................... left parenthesis 1 right parenthesis end style

Eqn.(1) is the required locus
Answered by Thiyagarajan K | 09 Jan, 2019, 08:27: AM

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