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)

$(7,4)$

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