Question

If V and S are respectively the vertex and focus of the parabola $y^2+6y+2x+5=0,$ then SV=

• A. 2
• B. $\frac{1}{2}$
• C. 1
• D. none of these

Answer 1

The correct option is B

$\frac{1}{2}$

Given:

The vertex and focus of the parabola are V and S,respectively

The equation of parabola can be rewritten as follows:

$(y+3{)}^2-9+5+2x=0$

$\Rightarrow (y+3{)}^2+2x=4$

$\Rightarrow (y+3{)}^2=4-2x$

$\Rightarrow (y+3{)}^2=-2(x-2)$

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

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

Vertex =(X=0,Y=0)=(x-2=0,y+3=0)=(x=2,y=-3)

Comparing with $y^2=4ax:$

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

Focus=$(X-\frac{1}{2},Y=0)$

=$(x-2=\frac{-1}{2},y+3=0)=((x=\frac{3}{2},y=-3)$

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