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Question

If the normal at the point P( $theta$ ) to the ellipse $fraction numerator x to the power of 2 end exponent over denominator 14 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 5 end fraction equals 1$ intersects it again at the point Q(2 $theta$ ) then cos $theta$ =

–2/3
2/3
– 6/7
6/7

hint Hint:

In this question, we have to find the value of cos $theta$ , if the normal at the point P( $theta$ ) to the ellipse $fraction numerator x to the power of 2 end exponent over denominator 14 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 5 end fraction equals 1$ intersects it again at the point Q(2 $theta$ ). First we will find the value of a and b using the equation given and then find the equation of normal passing through first point. Later substituting the second point in the first equation and simplify it to get the required value.

The correct answer is: –2/3

$x squared over 14 plus y squared over 5 equals 1 A s space g e n e r a l space e q u a t i o n space i s space x squared over a squared plus y squared over b squared equals 1 S o comma space a squared equals 14 rightwards double arrow a equals square root of 14 a n d space b squared equals 5 rightwards double arrow b equals square root of 5 N o w comma space e q u a t i o n space o f space n o r m a l space p a s sin n g space t h r o u g h space p o i n t space P left parenthesis theta right parenthesis space i s space g i v e n space b y fraction numerator a x over denominator cos theta end fraction minus fraction numerator b y over denominator sin theta end fraction equals a squared minus b squared rightwards double arrow fraction numerator square root of 14 x over denominator cos theta end fraction minus fraction numerator square root of 5 y over denominator sin theta end fraction equals 14 minus 5 rightwards double arrow fraction numerator square root of 14 x over denominator cos theta end fraction minus fraction numerator square root of 5 y over denominator sin theta end fraction equals 9 space space space space space _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ e q u a t i o n space i A g a i n space n o r m a l space m e e t space t h e space c u r v e space a t space Q left parenthesis 2 theta right parenthesis space i. e. comma space open parentheses square root of 14 cos space 2 theta comma square root of 5 sin space 2 theta space close parentheses N o w comma space p u t t i n g space i t space a t space e q u a t i o n space i comma space space space space fraction numerator 14 cos space 2 theta over denominator cos space theta end fraction minus fraction numerator 5 sin space 2 theta over denominator sin space theta end fraction equals 9 rightwards double arrow 14 open parentheses 2 space cos squared theta minus 1 close parentheses fraction numerator 1 over denominator cos space theta end fraction minus fraction numerator 10 sin space theta. cos space theta over denominator sin space theta end fraction equals 9 rightwards double arrow open parentheses 28 space cos squared theta space minus 14 close parentheses space fraction numerator 1 over denominator cos space theta end fraction minus 10 space cos space theta equals 9 rightwards double arrow fraction numerator 28 space cos squared theta space minus 14 minus 10 space cos squared space theta over denominator cos space theta end fraction equals 9 rightwards double arrow 18 space cos squared theta space minus 14 equals 9 cos space theta rightwards double arrow 18 space cos squared theta space minus 9 space cos space theta minus 14 equals 0 rightwards double arrow 18 space cos squared theta space minus 21 space cos space theta plus 12 cos space theta minus 14 equals 0 rightwards double arrow 3 cos space theta left parenthesis 6 cos space theta minus 7 right parenthesis plus 2 left parenthesis 6 cos space theta minus 7 right parenthesis equals 0 rightwards double arrow left parenthesis 6 cos space theta minus 7 right parenthesis open parentheses 3 cos space theta plus 2 close parentheses equals 0 rightwards double arrow cos space theta equals 7 over 6 space left parenthesis n o t space p o s s i b l e right parenthesis space a n d space cos space theta equals fraction numerator negative 2 over denominator 3 end fraction S o comma space cos space theta equals fraction numerator negative 2 over denominator 3 end fraction$

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