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The axis of a parabola is along the line y = x and the distance of its vertex A from 0 , 0 is 2 and that of its focus S from 0 , 0 is 2 2 . If A and S lie in first quadrant, then the equation of the parabola in parametric form is

Options

  1. Ax = t + 1 2 , y = t - 1 2
  2. Bx = t 2 , y = 2 t
  3. Cx = t - 2 2 , y = t + 2 2
  4. Dx = t 2 + 5 , y = t 2 - 5

Correct answer

A. x = t + 1 2 , y = t - 1 2

Step-by-step solution

Distance of vertex from origin = 2 Distance of focus from origin = 2 2 Axis is along y = x . The focus and the vertex lie in the first quadrant. Hence, the coordinates of the focus are 2 , 2 . The vertex is at 1 , 1 Hence, the equation of directrix is x + y = 0 Take point h , k on the parabola. By definition of the parabola h - 2 2 + k - 2 2 = h + k 2 2 ⇒ 2 h 2 + 2 k 2 + 16 - 8 h - 8 k = h 2 + k 2 + 2 h k ⇒ h 2 + k 2 - 8 h - 8 k - 2 h k + 16 = 0 Substitute h , k to x , y x 2 + y 2 - 2 x y - 8 x - 8 y +

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