If the tangent to the curve y = x ^ 3 at the point P(t, t ^ 3 )...

$= {\left(3{x}^{2}\right)}_{x = t} = 3{t}^{2}$

So equation tangent at $P\left(t, {t}^{3}\right):$

$y - {t}^{3} = 3{t}^{2}\left(x - t\right)$
for point of intersection with $y = {x}^{3}$

${x}^{3} - {t}^{3} = 3{t}^{2}x - 3{t}^{3}$

$⇒ \left(x - t\right)\left({x}^{2} + xt + {t}^{2}\right) = 3{t}^{2}\left(x - t\right)$

for $x \ne t$

${x}^{2} + xt + {t}^{2} = 3{t}^{2}$

$⇒ {x}^{2} + xt - 2{t}^{2} = 0 ⇒ \left(x - t\right)\left(x + 2t\right) = 0$

So for $Q: x = - 2t, Q\left(- 2t, - 8{t}^{3}\right)$
ordinate of required point $: \dfrac{2{t}^{3} - 8{t}^{3}}{2 + 1} = - 2{t}^{3}$