Question

Let O be the vertex, Q be any point on the parabola x²=8y. If point P divides the line segment OQ internally in the ratio 1:3, then the locus of P is 

• A. x²=y
• B. y²=x
• C. y²=2x
• D. x²=2y

Answer 1

The Correct Option is D

We have a parabola with the equation x^2 = 8y. Any point on this parabola can be represented as (4t, 2t^2), where t is a parameter.

The problem states that there's a point P that divides the line segment joining the origin O(0,0) and the point Q(4t, 2t^2) in the ratio 1:3. We're going to apply the section formula for internal division to find the coordinates of point P.

Let's call the coordinates of point P (h, k). We want to find the values of h and k.

By the section formula for internal division, the x-coordinate (h) of the point P is given by:

h = (1 * 4t + 3 * 0) / (1 + 3) = 4t/4 = t

The y-coordinate (k) of the point P is given by:

k = (1 * 2t^2 + 3 * 0) / (1 + 3) = 2t^2/4 = t^2/2

Now, we want to relate the x-coordinate and y-coordinate of point P using the given equation of the parabola x^2 = 8y.

Substituting h (which is t) and k (which is t^2/2) into the equation x^2 = 8y, we get:

(t)^2 = 8 * (t^2/2)

Simplifying, we find:

t^2 = 4t^2

Dividing both sides by t^2 (since t ≠ 0), we get:

1 = 4

The correct answer is option (D): x²=2y