If V and S are respectively the vertex and focus of the parabola … - Vidyadip

MCQ

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

A
$2$
✓
$\frac{1}{2}$
C
$1$
D
None of these

✓

Answer

Correct option: B.

$\frac{1}{2}$

$\frac{1}{2}$

Solution:
Given:
The vertex and the focus of a parabola are V and S, respectively.
The given equation of parabola can be rewritten as follows:
$(y+3)^2-9+5+2 x=0$
$\Rightarrow(y+3)^2+2 x=4$
$\Rightarrow(y+3)^2=4-2 x$
$\Rightarrow(y+3)^2=-2(x-2)$
$\text { Let } Y=y+3, X=x-2$
Then, the equation of parabola becomes $Y^2=-2 X$.
$\text { Vertex }=(X=0, Y=0)=(x-2=0, y+3=0)=(x=2, y=-3)$
Comparing with $y^2=4 a x$ :
$4 \mathrm{a}=2 \Rightarrow \mathrm{a}=\frac{1}{2}$
$\text { Focus }=\left(\mathrm{X}=\frac{-1}{2}, \mathrm{Y}=0\right)=\left(\mathrm{x}-2=\frac{-1}{2}, \mathrm{y}+3=0\right)=\left(\mathrm{x}=\frac{3}{2}, \mathrm{y}=-3\right)$
$\Rightarrow \mathrm{SV}=\sqrt{\left(2-\frac{3}{2}\right)^2+(-3+3)^2}=\frac{1}{2}$

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