Question

Let the tangent at point $P(4,8)$ to the curve $y^2=x^3$ meets the curve again at $Q.$ Then the length of $PQ$ is

• A. $\sqrt{10}$
• B. $2\sqrt{10}$
• C. $3\sqrt{10}$
• D. $4\sqrt{10}$

Answer 1

The correct option is C $3\sqrt{10}$

Given curve : $y^2=x^3$
Let point $Q=(t^2,t^3)$
For slope of tangent at $P:2y\frac{dy}{dx}=3x^2$
So, slope of tangent at $P(4,8)=$ slope of $PQ$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{(4,8)}=\frac{t^3-8}{t^2-4}$
$\Rightarrow 3=\frac{t^3-8}{t^2-4}$
$\Rightarrow 3=\frac{(t-2)(t^2+2t+4)}{(t-2)(t+2)}$
$\Rightarrow t^2-t-2=0$
$\Rightarrow t=-1,2;(t\ne 2)$
So, $Q\equiv (1,-1)$
$\Rightarrow PQ=\sqrt{81+9}=3\sqrt{10}$