Question

If V and S are respectively the vertex and focus of the parabola y² + 6y + 2x + 5 = 0, then SV =
(a) 2
(b) 1/2
(c) 1
(d) none of these

Answer 1

(b) 1/2

Given:
The vertex and the focus of a parabola are V and S, respectively.

The given equation of parabola can be rewritten as follows:
${\left(y+3\right)}^2-9+5+2x=0$
$$\begin{gathered} \Rightarrow {\left(y+3\right)}^2+2x=4 \\ \Rightarrow {\left(y+3\right)}^2=4-2x \\ \Rightarrow {\left(y+3\right)}^2=-2\left(x-2\right) \end{gathered}$$

Let $Y=y+3,X=x-2$

Then, the equation of parabola becomes $Y^2=-2X$.

Vertex = $\left(X=0,Y=0\right)=\left(x-2=0,y+3=0\right)=\left(x=2,y=-3\right)$

Comparing with y² = 4ax:

$4a=2\Rightarrow a=\frac{1}{2}$

Focus = $\left(X=\frac{-1}{2},Y=0\right)=\left(x-2=\frac{-1}{2},y+3=0\right)=\left(x=\frac{3}{2},y=-3\right)$

⇒ SV = $\sqrt{{\left(2-\frac{3}{2}\right)}^2+{\left(-3+3\right)}^2}=\frac{1}{2}$