Question

The equation of the parabola whose focus is at $(-1,-2)$ and the directrix is the line $x-2y+3=0$

• A. $4x^2+y^2+4xy+4x+32y+16=0$
• B. $4x^2+y^2+4xy+2x+8y+16=0$
• C. $4x^2+y^2+4xy+40x+16y+16=0$
• D. $4x^2+4y^2+4xy+4x+32y+16=0$

Answer 1

The correct option is A $4x^2+y^2+4xy+4x+32y+16=0$

Let $P(x,y)$ be any point on the parabola whose focus is $S(-1,-2)$ and the directrix $x-2y+3=0$.

Now, distance of $P$ from focus $=$ distance of $P$ from directrix.
$\Rightarrow \sqrt{(x+1{)}^2+(y+2{)}^2}=\frac{|x-2y+3|}{\sqrt{1+4}}$
$\Rightarrow (x+1{)}^2+(y+2{)}^2={\left(\frac{x-2y+3}{\sqrt{1+4}}\right)}^2$
$\Rightarrow 5\left[\right(x+1{)}^2+(y+2{)}^2]=(x-2y+3{)}^2\Rightarrow 5(x^2+y^2+1+2x+4y+4)=(x^2+4y^2+9-4xy+6x-12y)\Rightarrow 4x^2+y^2+4xy+4x+32y+16=0$

This is the equation of the required parabola.