Question

The coordinates of the focus of the parabola described parametrically by $x=5t^2+2$ and $y=10t+4$ are

• A. $(7,4)$
• B. $(3,4)$
• C. $(3,-4)$
• D. $(-7,4)$

Answer 1

The correct option is A $(7,4)$

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

$\therefore \frac{x-2}{5}={\left(\frac{y-4}{10}\right)}^2$
$\Rightarrow (y-4{)}^2=20(x-2)$
$\therefore$ Vertex $\equiv (2,4)$ and $a=5$
$\therefore$ Focus $\equiv (2+5,4)\equiv (7,4)$