Q.25. Parabola/Ellipse

Question 25.
Let P(x1, y1) and Q(x2, y2) where y1 < 0,y2 < 0 be the end points of the latus rectum of the ellipse x^2 + 4y^2 = 4, then the equation of the parabola with latus rectum PQ is...?

Options are
a)x^2+2*√(3) y = 3+√(3)                    b)x^2-2*√(3) y = 3+√(3)
c)x^2+2*√(3) y = 3-√(3)                     d)x^2-2*√(3) y = 3-√(3)

Answer 25.
x^2 + 4y^2 = 1

=> x^2/4 + y^2/1 = 1
Comparing with x^2/a^2 + y^2/b^2 = 1,
a^2 = 4 and b^2 = 1
This is a standard ellipse with major axis along x-axis and minor along y-axis.
e^2 = (a^2 - b^2)/a^2 = (4 - 1)/4 = √3/2
=> Foci are
S(ae, 0) = (√3, 0) and
S'(-ae, 0) = (-√3, 0)

As y1 < 0 and y2 < 0, P and Q are end-points of latus rectum passing through S and S' which are in the third and the fourth quadrants and are
P = (-ae, - b^2/a) = (-√3, - 1/2) and
Q = (ae, - b^2/a) = (√3, - 1/2).

As PQ is also a latus rectum of a parabola, its focus is (0, - 1/2)
and length of latus rectum is PQ = 2√3
=> eqn. of the parabola is
x^2 = 2√3 (y - c)
As (√3, - 1/2) lies on it,
3 = 2√3 (- 1/2 - c) => c = - 1/2 - √3/2
=> x^2 = 2√3 (y + 1/2 + √3/2)
=> x^2 - 2√3y = 3 + √3
This is answer b).

Also, eqn of parabola is
x^2 = - 2√3 (y - c)
As (√3, - 1/2) lies on it,
3 = - 2√3 (- 1/2 - c) => c = 1/2 + √3/2
=> x^2 = - 2√3 (y - 1/2 - √3/2)
=> x^2 + 2√3y = 3 + √3
Thus, answer a) is also possible.

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