Let $P Q$ be a double ordinate of the parabola, $y^{2} = - 4 x$ , where $P$ lies in the second quadrant. If $R$ divides $P Q$ in the ratio $2 : 1,$ then the locus of $R$ is

3 y^{2} = - 2 x

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9 y^{2} = 4 x

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9 y^{2} = - 4 x

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3 y^{2} = 2 x

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Solution

The correct option is B $9 y^{2} = - 4 x$
Let the coordinates of $P, Q$ and $R$ be $P (- {a t}_{1}^{2}, 2 {a t}_{1}), Q (- {a t}_{1}^{2}, - 2 {a t}_{1})$ and $R (h, k)$ .
As $R$ divides $P Q$ in the ratio $2 : 1$ then $h = \frac{- 1 {a t}_{1}^{2} - 2 {a t}_{1}^{2}}{2 + 1} = - {a t}_{1}^{2}$ ...(1)
and $k = \frac{1.2 {a t}_{1} - 2.2 {a t}_{1}}{2 + 1} = \frac{- 2 {a t}_{1}}{3}$ ...(2)
From (1) and (2) $9 k^{2} = - 4 h \Rightarrow 9 y^{2} = - 4 x$

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Let PQ be a double ordinate of the parabola, y2=−4x, where P lies in the second quadrant. If R divides PQ in the ratio 2:1, then the locus of R is

The correct option is B 9y2=−4xLet the coordinates of P,Q and R be P(−at21,2at1),Q(−at21,−2at1) and R(h,k).As R divides PQ in the ratio 2:1 then h=−1at21−2at212+1=−at21 ...(1)and k=1.2at1−2.2at12+1=−2at13 ...(2)From (1) and (2) 9k2=−4h⇒9y2=−4x

Let $P Q$ be a double ordinate of the parabola, $y^{2} = - 4 x$ , where $P$ lies in the second quadrant. If $R$ divides $P Q$ in the ratio $2 : 1,$ then the locus of $R$ is