Question

The focus of the parabola $x^2+2y+6x=0$ is :

• A. $(-3,4)$
• B. $(3,4)$
• C. $(3,-4)$
• D. $(-3,-4)$

Answer 1

Answer: (A)

Given that, $x^2+2y+6x=0$
$\Rightarrow x^2+6x=-2y...\left(i\right)$
On adding $9$ both sides of Eq. $\left(i\right)$, we get
$x^2+6x+9=-2y+9$
$\Rightarrow {\left(x+3\right)}^2=-2y+9$
Which is the form of parabola
$X^2=4aY$
where, $4a=-2$
$\Rightarrow a=-\frac{1}{2}$
$\therefore$ Focus of the parabola $X^2=4aY$ is $\left(0,a\right)$
$=\left(0,-\frac{1}{2}\right)$
But, $X=x+3$ and $Y=y-\frac{9}{2}$
$\therefore x=-3$ and $y=4$
[on putting $X=0$ and $Y=-1/2$]
$\therefore$ Focus of the parabola
$x^2+2y+6x=0$ is $(-3,4)$.