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 $
$\Rightarrow \frac{x-2}{5}=\left(\frac{y-4}{10}\right)^{2}$
$\Rightarrow \left(y-4\right)^{2}=20\left(x-2\right)$
$\Rightarrow y^{2}=20\,X$, 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 $\left(7,4 \right)$