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JEE Advanced2008MathematicsEllipseActual

Let P (x₁, y₁ ) and Q (x₂, y₂ ), y₁ < 0, y₂ < 0 , be the end points of the latusrectum of the ellipse x^2+4 y^2=4 . The equations of parabolas with latusrectum P Q are

Options

  1. Ax^2+2 3 y=3+ 3
  2. Bx^2-2 3 y=3+ 3
  3. Cx^2+2 3 y=3- 3
  4. Dx^2-2 3 y=3- 3

Correct answer

B. x^2-2 3 y=3+ 3

Step-by-step solution

The equation x^2+4 y^2=4 represents an ellipse with 2 and 1 as semi-major and semi-minor axes and eccentricity 3 2 . Thus, the ends of latusrect are ( 3 , 1 2 ) and ( 3 ,- 1 2 ), (- 3 ,- 1 2 ) and ( 3 ,- 1 2 ) . According to the question, we consider only P (- 3 ,- 1 2 ) and Q ( 3 ,- 1 2 ) . Now, P Q=2 3 Thus, the coordinates of the vertex of the parabolas are A (0, -1+ 3 2 ) and A^ (0, -1- 3 2 ) and corresponding equations are and (x-0)^2=-4 3 2 (y+ 1- 3 2 ) i.e., (x-0)^2=4 3 2 (y- -1- 3 2 ) and aligned & x^2+2 3

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