Locus of Point P on the Parabola x^2 = 8y
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Given that O is the vertex of the parabola and Q is any point on the parabola x^2 = 8y. We are also given that the point P divides the line segment OQ internally in the ratio 1:3.

To find the locus of point P, we need to determine the equation that represents all possible positions of P.

Using the concept of section formula, we can find the coordinates of point P. Let the coordinates of point O be (0,0) and the coordinates of point Q be (x, y).

Finding the coordinates of point P
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Since P divides the line segment OQ internally in the ratio 1:3, we can use the section formula to find the coordinates of P.

The section formula states that if a point P divides a line segment with endpoints (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of point P are given by:

Px = (mx2 + nx1) / (m + n)
Py = (my2 + ny1) / (m + n)

In this case, the coordinates of point O are (0,0) and the coordinates of point Q are (x, y). The ratio in which P divides OQ is 1:3, so m = 1 and n = 3.

Substituting the values into the section formula, we get:

Px = (3x + 1(0)) / (1 + 3) = (3x) / 4
Py = (3y + 1(0)) / (1 + 3) = (3y) / 4

Therefore, the coordinates of point P are (3x/4, 3y/4).

Finding the locus of point P
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To find the locus of point P, we need to eliminate x and y from the coordinates of P.

The equation of the locus can be found by substituting the coordinates of P into the equation x^2 = 8y.

Substituting (3x/4) for x and (3y/4) for y, we get:

(3x/4)^2 = 8(3y/4)
9x^2/16 = 24y/4
9x^2 = 96y
x^2 = 96y/9
x^2 = 32y/3

Therefore, the locus of point P is given by the equation x^2 = 32y/3, which is equivalent to x^2 = 2y.

Hence, the correct answer is option B, x^2 = 2y.