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Q.
If the tangent to the curve $y=x^{3}$ at the point $P \left( t , t ^{3}\right)$ meets the curve again at $Q ,$ then the ordinate of the point which divides $PQ$ internally in the ratio 1: 2 is :
Slope of tangent at $\left. P \left( t , t ^{3}\right)=\frac{ dy }{ dx }\right]_{\left( t , t ^{3}\right)}$
$=\left(3 x ^{2}\right)_{ x - t }=3 t ^{2}$
So equation tangent at $P \left( t , t ^{3}\right)$ :
$y-t^{3}=3 t^{2}(x-t)$
for point of intersection with $y=x^{3}$
$x^{3}-t^{3}=3 t^{2} x-3 t^{3}$
$\Rightarrow (x-t)\left(x^{2}+x t+t^{2}\right)=3 t^{2}(x-t)$
for $x \neq t$
$x^{2}+x t+t^{2}=3 t^{2}$
$\Rightarrow x^{2}+x t-2 t^{2}=0 $
$\Rightarrow (x-t)(x+2 t)=0$
So for $Q: x=-2 t, Q\left(-2 t,-8 t^{3}\right)$
ordinate of required point
$: \frac{2 t ^{3}-8 t ^{3}}{2+1}=-2 t ^{3}$