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

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(B) Let the focus of the parabola $y^2 = 12x$ be $S(3, 0)$. Let $P(3t^2, 6t)$ be any point on the parabola.Let $Q(h, k)$ be the point dividing $SP$ in

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$\frac{12m}{m+n}$

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(B) Let the focus of the parabola $y^2 = 12x$ be $S(3, 0)$. Let $P(3t^2, 6t)$ be any point on the parabola.Let $Q(h, k)$ be the point dividing $SP$ in the ratio $m:n$.Using the section formula,$h = \frac{m(3t^2) + n(3)}{m+n}$ and $k = \frac{m(6t) + n(0)}{m+n}$.From the second equation,$t = \frac{k(m+n)}{6m}$.Substitute $t$ into the equation for $h$:$h = \frac{3m(\frac{k(m+n)}{6m})^2 + 3n}{m+n} = \frac{3m \cdot \frac{k^2(m+n)^2}{36m^2} + 3n}{m+n} = \frac{\frac{k^2(m+n)^2}{12m} + 3n}{m+n}$.$h(m+n) = \frac{k^2(m+n)^2}{12m} + 3n$.$k^2(m+n)^2 = 12mh(m+n) - 36mn$.$k^2 = \frac{12m}{m+n}h - \frac{36mn}{(m+n)^2}$.This is of the form $Y^2 = 4AX$,where $4A = \frac{12m}{m+n}$.Thus,the length of the latus rectum is $\frac{12m}{m+n}$.

The locus of a point which divides the line segment joining the focus and any point on the parabola $y^2 = 12x$ in the ratio $m:n$ $(m+n \neq 0)$ is a parabola. Then the length of the latus rectum of that parabola is

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