Maths-
General
Easy

Question

If the chords of contact of P space open parentheses x subscript 1 comma y subscript 1 close parentheses and Q space open parentheses x subscript 2 comma y subscript 2 close parentheses w.r.t the ellipse fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1 are at right angle then fraction numerator x subscript 1 x subscript 2 over denominator y subscript 1 y subscript 2 end fraction equals

  1. fraction numerator a to the power of 2 end exponent over denominator text end text b to the power of 2 end exponent end fraction    
  2. fraction numerator negative b to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction    
  3. fraction numerator negative a to the power of 4 end exponent over denominator text end text b to the power of 4 end exponent end fraction    
  4. fraction numerator negative b to the power of 4 end exponent over denominator a to the power of 4 end exponent end fraction    

hintHint:

When two lines are perpendicular, the product of their slopes equals to -1.

The correct answer is: fraction numerator negative a to the power of 4 end exponent over denominator text end text b to the power of 4 end exponent end fraction


    E q u a t i o n space o f space c o n t a c t space o f space P open parentheses x subscript 1 comma y subscript 1 close parentheses space a n d space Q open parentheses x subscript 2 comma y subscript 2 close parentheses space a r e space g i v e n space b y colon
fraction numerator x x subscript 1 over denominator a squared end fraction plus fraction numerator y y subscript 1 over denominator b squared end fraction equals 1
fraction numerator x x subscript 2 over denominator a squared end fraction plus fraction numerator y y subscript 2 over denominator b squared end fraction equals 1

T h e space c h o r d s space a r e space p e r p e n d i c u l a r space t o space e a c h space o t h e r. space T h e r e f o r e comma space t h e i r space s l o p e s space w i l l space b e
m subscript 1 m subscript 2 equals negative 1
open parentheses x subscript 1 over a squared cross times b squared over y subscript 1 close parentheses open parentheses x subscript 2 over a squared cross times b squared over y subscript 2 close parentheses equals negative 1
open parentheses fraction numerator x subscript 1 x subscript 2 over denominator y subscript 1 y subscript 2 end fraction close parentheses open parentheses b to the power of 4 over a to the power of 4 close parentheses equals negative 1
therefore open parentheses fraction numerator x subscript 1 x subscript 2 over denominator y subscript 1 y subscript 2 end fraction close parentheses equals open parentheses fraction numerator negative a to the power of 4 over denominator b to the power of 4 end fraction close parentheses

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