Question

The locus of the mid-point of the line segment joining the focus of the parabola y² = 4ax to a moving point of the parabola, is another parabola whose directrix is ______.

Options

• x = a

• x = 0

• x = `-a/2`

• x = `a/2`

Answer 1

The locus of the mid-point of the line segment joining the focus of the parabola y² = 4ax to a moving point of the parabola, is another parabola whose directrix is x = 0.

Explanation:

Given: Parabola y² = 4ax

Focus of parabola, y² = 4ax is S(a, 0).

Consider a moving point P(at², 2at) on the parabola.

Let the midpoint of PS is Q(h, k)

Using the section formula,

h = `(a + at^2)/2`, k = `(0 + 2at)/2` = at

⇒ t²  = `(2h - a)/a` and t = `k/a`

∴ `k^2/a^2 = (2h - a)/a`

⇒ k² = `2a(h - a/2)`

Replace h by x and k by y to find locus of (h, k).

∴ Locus is y² = `2a(x - a/2)`

Let y = Y and `x - a/2` = X

∴ Locus become Y² = 2aX

Whose directrix is X = `(-a)/2`

⇒ `x = a/2 = (-a)/2`

⇒ x = 0

Hence, the equation of directrix is x = 0.