Question

Q is a point on (2y-1)^2=8(x-3). The locus of point on PQ dividing it in 2:1 ratio is? P is (-6,-1)

• y^2=12x
• y^2=4x
• None

Solution:

We will give 2 approaches both of which are pretty simple and yet need us to be careful.

Q(x_1,y_1) be a point on the given parabola. Now , let us denote the point obtained after applying section formula as (h,k).
Clearly,that point divides the line PQ in the ratio 2:1 so by applying the section formula we get.
h = (2x_1 - 6)/3 and k = (2y_1 - 1)/3

But we know (2y_1 - 1)^2 = 8(x_1 - 3)
Now by comparing and putting the values above in the parabola's equation, we get:
(3k)^2 = 8.(3h/2) k^2 = 4h/3 That is the equation of the locus of the required point.

Alternatively a naive approach would be to find the value of (x_1,y_1):
x_1 = (3h+6)/2 and y_1 = (3k+1)/2 Now putting the values of x_1 and y_1 in the equation of parabola, we get:
( 2(3k+1)/2 - 1)^2 = 8((3h+6)/2 - 3) 9k^2 = 12h k^2 = 4h/3