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Question

Equation of chord of the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ joining the points P (a cos $alpha$ , b sin $alpha$ ) and Q (a cos $beta$ , b sin $beta$ ) is 7

$\frac{x}{a} \cos (\frac{α + β}{2}) - \frac{y}{b} \sin (\frac{α + β}{2}) = \cos (\frac{α - β}{2})$
$\frac{x}{a} \cos (\frac{α + β}{2}) + \frac{y}{b} \sin (\frac{α + β}{2}) = \cos (\frac{α + β}{2})$
$\frac{x}{a} \cos (\frac{α + β}{2}) + \frac{y}{b} \sin (\frac{α + β}{2}) = \cos (\frac{α - β}{2})$
None of these above

hint Hint:

In this question, we have to find the correct equation if equation of chord of the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ joining the points P (a cos $alpha$ , b sin $alpha$ ) and Q (a cos $beta$ , b sin $beta$ ) is 7. First using points of the cord we will find the slope of the angle and later using that slope we can find the required equation.

The correct answer is: $fraction numerator x over denominator a end fraction cos invisible function application open parentheses fraction numerator alpha plus beta over denominator 2 end fraction close parentheses plus fraction numerator y over denominator b end fraction sin invisible function application open parentheses fraction numerator alpha plus beta over denominator 2 end fraction close parentheses equals cos invisible function application open parentheses fraction numerator alpha – beta over denominator 2 end fraction close parentheses$

$space space space space S l o p e space o f space theta equals fraction numerator b space sin alpha minus b space sin beta over denominator a space cos alpha minus a space cos beta end fraction equals b over a open parentheses fraction numerator 2 sin fraction numerator alpha minus beta over denominator 2 end fraction. cos fraction numerator alpha plus beta over denominator 2 end fraction over denominator negative 2 sin fraction numerator alpha plus beta over denominator 2 end fraction. sin fraction numerator alpha minus beta over denominator 2 end fraction end fraction close parentheses equals fraction numerator negative b over denominator a end fraction c o t fraction numerator alpha plus beta over denominator 2 end fraction space R e q u i r e d space e q u a t i o n comma space space space space space left parenthesis y minus b space sin space alpha right parenthesis equals fraction numerator negative b over denominator a end fraction c o t space fraction numerator alpha plus beta over denominator 2 end fraction. open parentheses x minus space a space cos space alpha close parentheses rightwards double arrow fraction numerator negative y over denominator b end fraction plus sin space alpha space equals 1 over a. c o t space fraction numerator alpha plus beta over denominator 2 end fraction. open parentheses x minus space a space cos space alpha close parentheses rightwards double arrow fraction numerator negative y over denominator b end fraction sin fraction numerator alpha plus beta over denominator 2 end fraction plus sin space alpha. sin space fraction numerator alpha plus beta over denominator 2 end fraction equals x over a cos fraction numerator alpha plus beta over denominator 2 end fraction minus cos fraction numerator alpha plus beta over denominator 2 end fraction cos alpha rightwards double arrow sin space alpha. sin fraction numerator alpha plus beta over denominator 2 end fraction plus cos space alpha. cos fraction numerator alpha plus beta over denominator 2 end fraction equals x over a cos fraction numerator alpha plus beta over denominator 2 end fraction plus y over b sin fraction numerator alpha plus beta over denominator 2 end fraction rightwards double arrow cos open parentheses alpha minus fraction numerator alpha plus beta over denominator 2 end fraction close parentheses equals x over a cos fraction numerator alpha plus beta over denominator 2 end fraction plus y over b sin fraction numerator alpha plus beta over denominator 2 end fraction rightwards double arrow x over a cos fraction numerator alpha plus beta over denominator 2 end fraction plus y over b sin fraction numerator alpha plus beta over denominator 2 end fraction equals cos open parentheses fraction numerator alpha minus beta over denominator 2 end fraction close parentheses$

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