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AP EAMCET202021 Sep 2020Evening ShiftMathematicsEllipseActual

Find the condition for the line (a x+b y+c=0 ) to be a normal to an ellipse ( x^2 4 + y^2 36 =1 )

Options

  1. A( 1 a^2 + 1 b^2 = 144 c^2 )
  2. B( 1 a^2 + 1 b^2 = 128 c^2 )
  3. C( 1 a^2 + 9 b^2 = 256 c^2 )
  4. D( 1 a^2 + 9 b^2 = 32 c^2 )

Correct answer

C. ( 1 a^2 + 9 b^2 = 256 c^2 )

Step-by-step solution

Let a point (P(2 , 6 ) ) on the ellipse ( x^2 4 + y^2 36 =1 ), so equation of normal to the ellipse at point (P ) is ( aligned & x-2 2 = y-6 6 & 2 x -4=6 y cosec -36 & 2 x -6 y cosec +32=0 (i) aligned ) Let normal (i) represent the line (a x+b y+c=0 ) ( aligned & So a 2 = b -6 cosec = c 32 & = c 16 a , =- 3 c 16 b & ^2 + ^2 =1 & c^2 256 a^2 + 9 c^2 256 b^2 =1 1 a^2 + 9 b^2 = 256 c^2 aligned ) Hence, option (c) is correct.

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