The locus of a point which divides the line s | vedclass

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(B) Let the point on the parabola $x^{2}=4y$ be $Q(2t, t^{2})$.Let the point $P(h, k)$ divide the line segment joining $A(0, -1)$ and $Q(2t, t^{2})$ i

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$9x^{2}-12y=8$

Vedclass · Answer

(B) Let the point on the parabola $x^{2}=4y$ be $Q(2t, t^{2})$.Let the point $P(h, k)$ divide the line segment joining $A(0, -1)$ and $Q(2t, t^{2})$ in the ratio $1:2$.Using the section formula,the coordinates of $P$ are:$h = \frac{1(2t) + 2(0)}{1+2} = \frac{2t}{3} \Rightarrow t = \frac{3h}{2}$$k = \frac{1(t^{2}) + 2(-1)}{1+2} = \frac{t^{2}-2}{3} \Rightarrow 3k = t^{2}-2$Substituting $t = \frac{3h}{2}$ into the equation $3k = t^{2}-2$:$3k = \left(\frac{3h}{2}\right)^{2} - 2$$3k = \frac{9h^{2}}{4} - 2$$12k = 9h^{2} - 8$$9h^{2} - 12k = 8$Replacing $(h, k)$ with $(x, y)$,the locus is $9x^{2}-12y=8$.

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola $x^{2}=4y$ internally in the ratio $1:2$ is:

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