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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 $fraction numerator x subscript 1 x subscript 2 over denominator y subscript 1 y subscript 2 end fraction equals$

$\frac{a^{2}}{b^{2}}$
$\frac{- b^{2}}{a^{2}}$
$\frac{- a^{4}}{b^{4}}$
$\frac{- b^{4}}{a^{4}}$

hint Hint:

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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