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General
Easy

Question

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

  1. (2, –2)    
  2. (0, 0)    
  3. (–1, 1)    
  4. None of these    

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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