Question

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

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

Answer 1

Answer: (A)

Given parametric curves are
$x=5t^2+2,y=10t+4$
or $\frac{x-2}{5}=t^2,\frac{y-4}{10}=t$
$\Rightarrow \frac{x-2}{5}={\left(\frac{y-4}{10}\right)}^2$
$\Rightarrow {\left(y-4\right)}^2=20\left(x-2\right)$
$\Rightarrow y^2=20X$, where $y=y-4$, $X=x-2$
$\therefore$ Coordinates of focus are $\left(5,0\right)$
i.e, $x-2=5$,
$\Rightarrow x=7$
$y-4=0$
$\Rightarrow y=4$
Hence, required coordinates are $\left(7,4\right)$