Let O be the vertex and Q be any point on the parabola,`x^2=""8y` . It the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is : (1) `x^2=""y` (2) `y^2=""x` (3) `y^2=""2x` (4) `x^2=""2y`

Explanation

To find the locus of point P that divides the line segment OQ internally in the ratio 1:3, where O is the vertex of the parabola \(x^2 = 8y\) and Q is any point on the parabola, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Vertex and Point Q**:
- The vertex O of the parabola \(x^2 = 8y\) is at the origin, \(O(0, 0)\).
- Let the coordinates of point Q be \(Q(p, q)\), where \(Q\) lies on the parabola.

...