Question

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 :

• A. $-2t^3$
• B. 0
• C. $-t^3$
• D. $2t^3$

Answer 1

Answer: (A)

Slope of tangent at ${P\left(t,t^3\right)=\frac{dy}{dx}]}_{\left(t,t^3\right)}$
$={\left(3x^2\right)}_{x-t}=3t^2$
So equation tangent at $P\left(t,t^3\right)$ :
$y-t^3=3t^2(x-t)$
for point of intersection with $y=x^3$
$x^3-t^3=3t^{2}x-3t^3$
$\Rightarrow (x-t)\left(x^2+xt+t^2\right)=3t^2(x-t)$
for $x\ne t$
$x^2+xt+t^2=3t^2$
$\Rightarrow x^2+xt-2t^2=0$
$\Rightarrow (x-t)(x+2t)=0$
So for $Q:x=-2t,Q\left(-2t,-8t^3\right)$
ordinate of required point
$:\frac{2t^3-8t^3}{2+1}=-2t^3$