Maths-
General
Easy

Question

Let P be any point on any directrix of an ellipse Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at

  1. some point on the major axis depending upon the position of point P    
  2. the midpoint of the line segment joining the center to the corresponding focus    
  3. the corresponding focus    
  4. none of these    

The correct answer is: the corresponding focus


    Given that Let P be any point on any directrix of an ellipse Then we need to find the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at what point
    L e t space e l l i p s e space i s fraction numerator space x squared space space space over denominator a squared end fraction ​ plus space space y squared over b squared space ​ space space equals 1 space w h e r e space a greater than b greater than 0 space a n d space p o i n t space P space i s
left parenthesis space a over e space comma k right parenthesis
A u x i l i a r y space c i r c l e space i s space x squared plus y squared space equals a squared space space
C h o r d s space o f space c o n t a c t space a r e fraction numerator space x a over denominator e a squared end fraction plus fraction numerator y k over denominator b squared end fraction equals 1 space a n d fraction numerator space x a over denominator e end fraction plus y k equals a squared space space

open parentheses fraction numerator space x over denominator a e end fraction minus 1 close parentheses plus k open parentheses y over b squared close parentheses equals 0 space a n d space open parentheses fraction numerator space x a over denominator e end fraction minus a squared close parentheses equals plus k y equals 0

    which are family of concurrent lines passing through point (ae,0) i.e. focus of ellipse

    Therefore, the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at the corresponding focus.

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