Question

Let the hyperbola H : `x^2/a^2 - y^2/b^2` = 1 pass `(2sqrt(2), -2sqrt(2))`. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

विकल्प

• `(2sqrt(3), -3sqrt(2))`

• `(3sqrt(3), -6sqrt(2))`

• `(sqrt(3), -sqrt(6))`

• `(3sqrt(6), 6sqrt(2))`

Answer 1

`bb((3sqrt(3), -6sqrt(2))`

Explanation:

Given equation of hyperbola is

`x^2/a^2 - y^2/b^2` = 1

Foci: S(ae, 0), S'(–ae, 0)

∴ Foot of directrix of parabola is (–ae, 0)

Focus of parabola is (ae, 0)

Now, semi latus is rectum of parabola = |SS'| = 2ae

Given, 4ae = `e((2b^2)/a)`

⇒ b² = 2a²

Given `(2sqrt(2), -2sqrt(2))` lies on hyperbola

⇒ `1/a^2 - 1/b^2` = `1/8`  ...(ii)

Form (i) and (ii)

a² = 4, b² = 8;

∵ b² = a²(e² – 1)

∴ e = `sqrt(3)`

⇒ Equation of parabola is y² = `8sqrt(3)x`