Question

If the focus of a parabola is $(2,5)$ and the directrix is $y=3$, the equation of the parabola is

• A. $x^2+4x+4y+20=0$
• B. $x^2+4x-4y+20=0$
• C. $x^2-4x-4y+20=0$
• D. $x^2+4x+4y+10=0$

Answer 1

The correct option is C $x^2-4x-4y+20=0$

Let $(x,y)$ be any point on the parabola then the distance of this point from focus is equal to distance from directrix.

Distance of point from focus $=\sqrt{(x-2{)}^2+(y-5{)}^2}$
Distance of point from directrix $=\frac{|0.x+1.y-3|}{\sqrt{1^2+0^2}}$

$\therefore \sqrt{(x-2{)}^2+(y-5{)}^2}=|y-3|$
squaring both sides, we get
$\Rightarrow (x-2{)}^2+(y-5{)}^2=(y-3{)}^2$
$\Rightarrow x^2-4x+4+y^2-10y+25=y^2-6y+9$
$\Rightarrow x^2-4x-4y+20=0$