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 $y^2 + 6y + 2x + 5 = 0$, then SV =

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 $y^2=4 a x$ :
$4\text{a} = 2 \Rightarrow \text{a} =\frac{1}{2}$
Focus $=\ \Big(\text{X}=\frac{-1}{2}, \text{Y}=0\Big)=\Big(\text{x}-2=\frac{-1}{2},\text{y}+3=0\Big)=\Big(\text{x}=\frac{3}{2},\text{y}=-3\Big)$
$\Rightarrow\ \text{SV}=\sqrt{\Big(2-\frac{3}{2}\Big)^2+(-3+3)^2}=\frac{1}{2}$

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