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MHT CET2008MathematicsEllipse

If the line x +y =p be normal to the ellipse x² a² + y² b² =1 , then

Options

  1. Ap² (a² ² +b² ² )=a²-b²
  2. Bp² (a² ² +b² ² )= (a²-b² )²
  3. Cp² (a² ² +b² cosec ² )=a²-b²
  4. Dp² (a² ² +b² cosec ² )= (a²-b² )²

Correct answer

D. p² (a² ² +b² cosec ² )= (a²-b² )²

Step-by-step solution

The equation of any normal to x² a² + y² b² =1 is a x -b y cosec =a²-b² (i) The straight line x +y =p will be a normal to the ellipse x² a² + y² b² =1 , if Eq. (i) and x +y =p represent the same line. a = -b cosec = a²-b² p = a p (a²-b² ) , = -b p (a²-b² ) ² + ² =1 b² p² (a²-b² )² ² + a² p² (a²-b² )² ² =1 p² (b² cosec ² +a² ² )= (a²-b² )²

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