ParabolaHard
Question
The locus of the mid point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix
Options
A.x = -a
B.
C.x = 0
D.
Solution
Let P (h, k) be the mid point of the line segment joining the focus (a, 0) and a general point Q(x, y) on the parabola.
Then
⇒ x = ah - a, y = 2k
Put these values of x and y in y2 = 4ax, we get
4k2 = 4a(2h - a)
⇒ 4k2 = 8ah - 4a2 ⇒ k2 = 2ah - a2
So, locus of P ( h, k) is y2 = 2ax - a2
⇒ y2 = 2a
Its directrix is
⇒ x = 0
Then
⇒ x = ah - a, y = 2k
Put these values of x and y in y2 = 4ax, we get
4k2 = 4a(2h - a)
⇒ 4k2 = 8ah - 4a2 ⇒ k2 = 2ah - a2
So, locus of P ( h, k) is y2 = 2ax - a2
⇒ y2 = 2a
Its directrix is
⇒ x = 0
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