Question

Let O be the vertex and Q be any point on the parabola ${{x}^{2}}=8y$ . If the point P divides the line segment OQ internally in the ratio 1: 3, then the locus of P is
(a) ${{x}^{2}}=y$
(b) ${{y}^{2}}=x$
(c) ${{y}^{2}}=2x$
(d) ${{x}^{2}}=2y$

Answer 1

- Hint: Now, we know that the vertex of a general parabola (Which we can say by looking at the equation of the parabola) is at the origin that is (0, 0).
The section formula will also be used in solving this question that is as follows
$(x,y)=\left[ \left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{2} \right),\left( \dfrac{m{{y}_{2}}+n{{y}_{1}}}{2} \right) \right]$
(Where the point with coordinates (x, y) divides the line joining the points $({{x}_{1}},{{y}_{1}})\ and\ ({{x}_{2}},{{y}_{2}})$ in the ratio of m: n)

Complete step-by-step solution -

Now, for finding the locus of the point P, we will try to get the coordinates of point P in terms of known coordinates of the parabola.

As mentioned in the question, we have to find the locus of the point P which divides the line segment OQ internally in the ratio 1: 3.
Now, let the coordinates of the point P be (X, Y). We know that the point O is the origin or (0, 0). Now, we can take the coordinates of the point Q as $(\sqrt{8y},y)$ .
Now, we know that the point P divides the line segment OQ internally in the ratio 1: 3, so we can use the formula given in the hint as follows

& (X,Y)=\left[ \left( \dfrac{1\sqrt{8y}+3(0)}{1+3} \right),\left( \dfrac{1y+3(0)}{1+3} \right) \right] \\\ & (X,Y)=\left[ \left( \dfrac{\sqrt{8y}}{4} \right),\left( \dfrac{y}{4} \right) \right] \\\ \end{aligned}$$ Now, as the coordinates of the point P is given as (x, y), so we can write the coordinates of the point P can be written as follows $$X=\dfrac{\sqrt{8y}}{4},Y=\dfrac{y}{4}$$ Now, on comparing the values of X and Y, we can observe that we can equate the two as follows $$\begin{aligned} & {{X}^{2}}=\dfrac{8y}{16}=\dfrac{y}{2},Y=\dfrac{y}{4} \\\ & 2{{X}^{2}}=y,4Y=y \\\ & 2{{X}^{2}}=4Y \\\ & {{X}^{2}}=2Y \\\ \end{aligned}$$ _Hence, the correct option is (d)._ Note: -The students can make an error if they don’t know the section formula and also the procedure to solve the locus of a given point that is point P in this case. Without knowing these above mentioned formulas, one cannot get to the correct locus. Also, knowing the section formula for solving this question, becomes extremely crucial as without it, one can never reach the solution. The section formula will also be used in solving this question that is as follows $$(x,y)=\left[ \left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{2} \right),\left( \dfrac{m{{y}_{2}}+n{{y}_{1}}}{2} \right) \right]$$