Question

If the tangent at a point $P$, with parameter $t$, on the curve $x=4t^2+3,y=8t^3-1,t\in R$, meets the curve again at a point $Q$, then the coordinates of $Q$ are :

• A. $(t^2+3,-t^3-1)$
• B. $(4t^2+3,-8t^3-1)$
• C. $(t^2+3,t^3-1)$
• D. $(16t^2+3,-64t^3-1)$

Answer 1

Answer: (A)

$P\left({}^A{t}^2,38t^{3}1\right)$
$\frac{du/dt}{dx/dt}=\frac{dy}{dx}=3t$ (slope of tangent at P)
Let $Q=\left(4{\lambda }^2+3,8{\lambda }^3-1\right)$
slope of $PQ=3t$
$\frac{8t^3-8{\lambda }^3}{4t^2-4{\lambda }^2}=3t$
$\Rightarrow t^2+t\lambda -2{\lambda }^2=0$
$\left(t-\lambda \right)\left(t+2\lambda \right)=0$
$t=\lambda \left(or\right)\lambda =\frac{-t}{2}$
$\therefore Q=\left[t^2+3,t^3-1\right]$