Question

A circle is drawn through the extremities of the latus rectum of the parabola $y^2=20x$. If the circle touches the directrix of the parabola, then the radius of the circle is(units)

Answer 1

Given parabola is $y^2=20x$
Extremities of the latus rectum of the parabola are
$(5,10)$ and $(5,-10)$
Let the center of the circle be $(h,k)$ and radius is $r$, then
$(x-h{)}^2+(y-k{)}^2=r^2$
This circle passes through $(5,10)$ and $(5,-10)$, then
$(5-h{)}^2+(10-k{)}^2=r^2\cdots \left(1\right)(5-h{)}^2+(-10-k{)}^2=r^2\cdots \left(2\right)$
From equation $\left(1\right)$ and $\left(2\right)$, we get
$k=0$
$(5-h{)}^2+100=r^2\cdots (3)$
The circle touches the directrix $x=-5$
$\frac{|h+5|}{1}=r\Rightarrow (h+5{)}^2=r^2$
Using equation $\left(3\right)$, we get
$\Rightarrow (h+5{)}^2=(5-h{)}^2+100\Rightarrow h=5\therefore r=10\text{units}$