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 \quad...\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 $(-3, 4)$.