The coordinates of the focus of the parabola described parametrically by x = 5 t 2 + 2 , y = 10 t + 4 (where…

Given parametric curves are

$x=5t^2+2, y=10t+4$

$\Rightarrow \frac{x-2}{5}=t^2,\frac{y-4}{10}=t$

Eliminating $t$, we get

$\frac{x-2}{5}={\left(\frac{y-4}{10}\right)}^2$

$\Rightarrow {\left(y-4\right)}^2=20\left(x-2\right)$

$\Rightarrow Y^2=10X,$ where $Y=y-4, X=x-2$

Coordinates of focus are $\left(5,0\right)$

So, $x-2=5, y-4=0$

$\Rightarrow x=7, y=4$

Hence, required coordinates are $\left(7,4\right)$