Parametic equation of a curve are given X=at² and y=at³ where a is a constant. Find Equation of tangent at point where t=2. Hence find coordinates where tangent meets curve again.

Question

Parametic equation of a curve are given X=at²  and y=at³ where a is a constant. Find Equation of tangent at point where t=2. Hence find coordinates where tangent meets curve again.

anonymous · Answer

$$x^3=a^3t^6 \implies y^2=a^2t^6$$
$$x^3=ay^2$$
$$2a y y'=3x^2$$
when $$t=2,x=2^6a^3,y^2=a^22^6$$
find y'

anonymous · Answer

$$y'=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{2at}{3at^2}=\frac{2}{3}t~\text{ for }~t\not=0$$
Slope of tangent line at $t=2$:
$$y'(2)=\frac{2}{3}\times 2=\frac{4}{3}$$
Point on curve when $t=2$:
$$\begin{cases}x(2)=a\times2^2\y(2)=a\times2^3\end{cases}~\Rightarrow~(x,y)=(4a,8a)$$
So now find the equation of the tangent line that passes through $(4a,8a)$ with slope $\dfrac{4}{3}$, then find the intersection with the curve for $t\not=2$.