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Question

If CP and CD are semi – conjugate diameters of an 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 comma then CP2 + CD2 =

  1. a + b    
  2. a2 + b2    
  3. a2 – b2    
  4. square root of a to the power of 2 end exponent plus b to the power of 2 end exponent end root    

Hint:

Properties of conjugate diameters
i)Sum of squares of any two conjugate semi diameters of ellipse is equal to sum of square of semi axes of ellipse.
ii) Square of semi diameter which is conjugate to the diameter through a point is equal to the product of focal distances of a point on an ellipse.

The correct answer is: a2 + b2


    Given : CP and CD are semi – conjugate diameters of an 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 comma
    Let coordinates of P(a costheta , b sintheta)
    Coordinates of D( a cosleft parenthesis theta space plus straight pi over 2 right parenthesis , a sinleft parenthesis theta space plus space straight pi over 2 right parenthesis) = (-a sintheta , b costheta)
    rightwards double arrow C P squared space plus space C D squared space equals space a squared cos squared theta space plus space b squared sin squared theta space plus space a squared sin squared theta space plus space b squared cos squared theta
rightwards double arrow C P squared space plus space C D squared space equals space a squared left parenthesis cos squared theta space plus sin squared theta right parenthesis space plus space b squared left parenthesis sin squared theta space plus space cos squared theta right parenthesis
rightwards double arrow C P squared space plus space C D space equals space a squared space plus space b squared....... space left parenthesis cos squared theta space plus space sin squared theta space equals space 1 right parenthesis

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