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

Given: Parabola with equation: \[ x^2 = 8y \] Any point on this parabola can be represented as: \[ Q = (4t, 2t^2) \]

Let O = (0, 0) and point P divides the line segment OQ in the ratio 1:3 internally.

Using section formula:
Coordinates of P (h, k) are: \[ h = \frac{1 \cdot 4t + 3 \cdot 0}{1 + 3} = \frac{4t}{4} = t \] \[ k = \frac{1 \cdot 2t^2 + 3 \cdot 0}{1 + 3} = \frac{2t^2}{4} = \frac{t^2}{2} \] So, P = (t, t^2/2)

Now eliminate parameter t: 
From \( h = t \), we get \( t = h \).
Substitute into k: \[ k = \frac{h^2}{2} \]

Hence, the locus of point P is: \[ y = \frac{x^2}{2} \Rightarrow x^2 = 2y \]

Final Answer: Option (D): \( x^2 = 2y \)