Question

Let $m_1$ be the slope of the tangent at a point $P$ on the curve $y=x^3.$ This tangent at $P$ intersects the curve again at $Q.$ If the slope of tangent at $Q$ is $m_2$ and $m_2=k{m}_1,$ then the value of $k$ is

Answer 1

Let $P\equiv (x_1,y_1),Q\equiv (x_2,y_2)$
$y=x^3\frac{dy}{dx}=3x^2\Rightarrow m_1=3x_1^2\cdots \left(1\right)$
Equation of tangent at $P(x_1,y_1)$ is
$y-y_1=3x_1^2(x-x_1)\Rightarrow y-x_1^3=3x_1^2(x-x_1)\Rightarrow y-3x_1^{2}x+2x_1^3=0\cdots \left(2\right)$

Tangent at $P$ intersects the curve again at $Q(x_2,y_2)$
So, substituting $(x_2,x_2^3)$ in eqn$\left(2\right)$, we get
$x_2^3-3x_1^2{x}_2+2x_1^3=0\Rightarrow (x_2-x_1)(x_2^2+x_1{x}_2+x_1^2-3x_1^2)=0$
$\Rightarrow x_2^2+x_1{x}_2-2x_1^2=0(\because x_1\ne x_2)$
$\Rightarrow (x_2-x_1)(x_2+2x_1)=0$
$\Rightarrow x_2=-2x_1(\because x_1\ne x_2)$
Now, slope of tangent at $Q(x_2,y_2)$ is
$m_2=3x_2^2=3(-2x_1{)}^2=4(3x_1^2)\Rightarrow k=4$