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Question

If P (q) and Q $open parentheses fraction numerator pi over denominator 2 end fraction plus theta close parentheses$ are two points on 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 locus of the mid – point of PQ is

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{1}{2}$
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 4$
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 2$
none of these

hint Hint:

Equation of ellipse = $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$

The correct answer is: $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals fraction numerator 1 over denominator 2 end fraction$

Coordinate of Point P( $theta$ ) = (a cos $theta$ , b sin $theta$ )
Coordinate of Point Q ( $straight pi over 2 space plus space theta$ ) = ( a cos( $straight pi over 2 space plus space theta$ ) , $b space sin left parenthesis straight pi over 2 space plus space theta right parenthesis$ )
Coordinate of Point Q ( $straight pi over 2 space plus space theta$ ) = ( -a sin( $space theta$ ) , $b space cos left parenthesis space theta right parenthesis$ )
Let R be the midpoint of P and Q and coordinates of R = (h, k)
$h space equals space fraction numerator a cos theta space minus space a sin theta over denominator 2 end fraction space a n d space k space equals space fraction numerator b sin theta space plus space b cos theta over denominator 2 end fraction$
$fraction numerator 2 h over denominator a end fraction space equals space cos theta space minus sin theta space a n d space fraction numerator 2 k over denominator b end fraction space equals space sin theta space plus space cos theta$
$left parenthesis fraction numerator 2 h over denominator a end fraction right parenthesis squared space plus space left parenthesis fraction numerator 2 k over denominator b end fraction right parenthesis squared space equals space open parentheses cos theta space minus sin theta close parentheses squared space plus left parenthesis cos theta space plus sin theta right parenthesis squared$
$left parenthesis fraction numerator 4 h squared over denominator a squared end fraction right parenthesis space plus space left parenthesis fraction numerator 4 k squared over denominator b squared end fraction right parenthesis space equals space open parentheses cos squared theta space plus sin squared theta space minus space 2 cos theta sin theta close parentheses space plus left parenthesis cos squared theta space plus sin squared theta space plus space 2 cos theta sin theta right parenthesis$
$left parenthesis fraction numerator 4 h squared over denominator a squared end fraction right parenthesis space plus space left parenthesis fraction numerator 4 k squared over denominator b squared end fraction right parenthesis space equals space 2$
$left parenthesis fraction numerator 2 h squared over denominator a squared end fraction right parenthesis space plus space left parenthesis fraction numerator 2 k squared over denominator b squared end fraction right parenthesis space equals 1$ comparing with equation of ellipse i.e. $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$
$x squared over a squared plus y squared over b squared equals 1 half$

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