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Question

If x cos $alpha$ + y sin $alpha$ = p is a tangent to the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , then -

$a^{2} {s i n}^{2} α + b^{2} {c o s}^{2} α = p^{2}$
$a^{2} + b^{2} {s i n}^{2} α = p^{2} {c o s e c}^{2} α$
$a^{2} {c o s}^{2} α + b^{2} {s i n}^{2} α = p^{2}$
None of these

hint Hint:

In this question, we have to find the correct option if the line x cos $alpha$ + y sin $alpha$ = p touches the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ . For this we will first have to solve the given equation to find the value of $x subscript 1 space a n d space y subscript 1$ and later substitute in the equation of ellipse to find the correct equation.

The correct answer is: $a to the power of 2 end exponent c o s to the power of 2 end exponent invisible function application alpha plus b to the power of 2 end exponent s i n to the power of 2 end exponent invisible function application alpha equals p to the power of 2 end exponent$

$space space space space space x space cos space alpha space plus space y space sin space alpha space equals p rightwards double arrow fraction numerator x space cos space alpha over denominator p end fraction plus fraction numerator y space sin space alpha over denominator p end fraction equals 1 rightwards double arrow x subscript 1 over a squared equals fraction numerator cos space alpha over denominator p end fraction comma space y subscript 1 over b squared equals fraction numerator sin space alpha over denominator p end fraction rightwards double arrow x subscript 1 equals fraction numerator cos space alpha space a squared over denominator p end fraction comma space y subscript 1 equals fraction numerator sin space alpha space b squared over denominator p end fraction A s space t h e r e space a r e space t w o space p o i n t s space o n space t h e space e q u a t i o n space o f space s t r a i g h t space l i n e space S o comma space fraction numerator a to the power of 4 space cos space alpha space over denominator a squared p end fraction plus fraction numerator b to the power of 4 space sin space alpha space over denominator b squared p end fraction equals p rightwards double arrow space fraction numerator a squared space cos space alpha space over denominator p end fraction plus fraction numerator b squared space sin space alpha space over denominator p end fraction equals p rightwards double arrow a squared cos squared space alpha plus b squared sin squared space alpha equals p squared$

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