Maths-
General
Easy

Question

If the normal at the point straight P left parenthesis theta right parenthesis to the ellipse x squared over 14 plus y squared over 5 equals 1 intersects it again at the point straight Q left parenthesis 2 theta right parenthesis, then cos space theta is

  1. 2/3
  2. -2/3
  3. 3/4
  4. none of these.

Hint:

In this question we will write the parametric equation of points on the ellipse using the given angle, then we will find their slope also we will find the equation of the normal in the parametric form. Then we will equate the two slopes and simplify the equation using trigonometric identities and find the required trigonometric ratio.

The correct answer is: -2/3


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

 

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