Parametric equations - finding A

Question

The curve with parametric equations x = a(t-2), y = at² + 2 (where a≠0), meets the y-axis at the point (0,5).

(a) Find the value of the constant a.

(b) Hence determine whether the curve meets the x-axis

Answer 1

For part (a) we have \begin{align} \big(\,x,\,y\,\big)=\big(\,0,\,5\,\big)\implies \begin{cases} a\,t-2\,a=0 \\ a\,t^2+2 = 5 \end{cases} \implies \begin{cases} a\,t =2\,a \\ a\,t^2 = 3 \end{cases} \implies \begin{cases} t =2 \\ a = \dfrac{3}{4} \end{cases} \end{align} For part (b) observe that $$ y = \dfrac{3}{4}\,t^2+2 \implies \forall \; t \in \mathbb R \quad y\left(t\right)\ge 2$$ Therefore curve never reaches $\,x\,$ axis.