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Prove that locus of a point whose chord of contact with respect to parabola passes through focus is directrix

Prove that locus of a point whose chord of contact with respect to parabola passes through focus is directrix

Grade:11

1 Answers

Partha Math Expert - askIITians
askIITians Faculty 25 Points
3 years ago
Let the parabola be y2= 4ax and any ponit on the locus be (h,k). Then the chord of contact of (h,k) is given by:
T = 0 or 2yk – 4a(x+h) = 0
Since it passes through the focus (a,0).
Plugging it in: a + h = 0
Thus, x = -a is the locus, which is also the directrix.

Here, T = 0 means doing the following subsitutions:
x2→ xh
y2→ yk
x → (x + h)/2
y → (y+k)/2
xy → (xk + yh)/2
in the general equation of any conic, which is y2– 4ax = 0 in this case.

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