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Maths-

General

Easy

Question

The normal to the curve x = 3 cos $theta$ – $cos cubed theta$ , y = 3 sin $theta$ – $sin cubed$ $theta$ at the point $theta$ = $pi$ /4 passes through the point -

(2, –2)
(0, 0)
(–1, 1)
None of these

hint Hint:

Find, Slope of tangent = $fraction numerator d y over denominator d x end fraction$ = $fraction numerator begin display style bevelled fraction numerator d y over denominator d theta end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d theta end fraction end style end fraction$
Then find slope of normal, and substitute the values in equation of normal line.

The correct answer is: (0, 0)

Given :
x = $3 cos theta space minus space cos cubed theta$
$y space equals space 2 sin theta space minus space sin cubed theta$
At $theta space equals space straight pi over 4$
$x space equals space 3 cos straight pi over 4 space minus space space cos cubed straight pi over 4 space equals space 3 cross times fraction numerator 1 over denominator square root of 2 end fraction space minus space fraction numerator 1 over denominator 2 square root of 2 end fraction space equals space fraction numerator 5 over denominator 2 square root of 2 end fraction y space equals space 2 sin straight pi over 4 space minus space sin cubed straight pi over 4 space equals space 2 cross times fraction numerator 1 over denominator square root of 2 end fraction space minus space fraction numerator 1 over denominator 2 square root of 2 end fraction space equals space fraction numerator 5 over denominator 2 square root of 2 end fraction$
At P( $theta space equals space straight pi over 4 right parenthesis$ = $left parenthesis fraction numerator 5 over denominator 2 square root of 2 end fraction comma space fraction numerator 5 over denominator 2 square root of 2 end fraction right parenthesis$
Slope of tangent = $fraction numerator d y over denominator d x end fraction$ = $fraction numerator begin display style bevelled fraction numerator d y over denominator d theta end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d theta end fraction end style end fraction$
$fraction numerator d y over denominator d theta end fraction space equals space 3 cos theta space minus space 3 sin squared theta cos theta fraction numerator d x over denominator d theta end fraction space equals space minus 3 sin theta space plus thin space space 3 cos squared theta sin theta$
$fraction numerator d y over denominator d x end fraction$ = $fraction numerator begin display style bevelled fraction numerator d y over denominator d theta end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d theta end fraction end style end fraction space equals space fraction numerator 3 cos theta space minus space 3 sin squared theta cos theta over denominator negative 3 sin theta space plus space 3 cos squared theta sin theta end fraction$
Slope of tangent at point P( $theta space equals space straight pi over 4 right parenthesis$ = $fraction numerator 3 cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style space minus space 3 cross times begin display style 1 half end style cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator negative 3 cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style space plus space 3 cross times begin display style 1 half end style space cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction space equals space fraction numerator begin display style bevelled fraction numerator 3 over denominator 2 square root of 2 end fraction end style over denominator begin display style bevelled fraction numerator negative 3 over denominator 2 square root of 2 end fraction end style end fraction space equals space minus 1$

When slope of tangent at P= -1
Slope of normal at point P = 1
Equation of normal at point P
y - y1 = m(x-x1)
$y space minus space fraction numerator 5 over denominator 2 square root of 2 end fraction space equals space 1 open parentheses x minus space fraction numerator 5 over denominator 2 square root of 2 end fraction close parentheses space rightwards double arrow space y space equals space x W h e n space y space equals space x comma space t h i s space m e a n s space t h e space l i n e space p a s s e s space t h r o u g h space t h e space o r i g i n space left parenthesis 0 comma 0 right parenthesis$

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