Question

For the parabola $y^2+6y-2x+5=0$, match the items in List-I with the suitable item in List-II given below:

	List-I		List-II
(I)	Vertex	(A)	$\left(-\frac{3}{2},-3\right)$
(II)	Focus	(B)	$\left(\frac{3}{2},-3\right)$
(III)	Equation of the directrix	(C)	$2x+5=0$
(IV)	Equation of the axis	(D)	$2x+y+3=0$
		(E)	$y+3=0$
		(F)	$(-2,-3)$

The correct matching isI

• A. I -F , II - A , III -E , IV -C
• B. I - F , II - A , III -C , IV -E
• C. I -A , II - B , III - C , IV - D
• D. I -F , II -A , III -C , IV - D

Answer 1

Answer: (B)

Parabola, $y^2+6y-2x+5=0$
$\Rightarrow y^2+6y=2x-5$
$\Rightarrow y^2+6y+9=2x-5+9$
$\Rightarrow (y+3{)}^2=2(x+2)$
$\Rightarrow Y^2=2X$
$($ where, $Y=y+3$ and $X=x+2)$
This parabola is in the form of $y^2=4ax$ by comparing
$\Rightarrow 4a=2$
$a=\frac{1}{2}$
Vertex $=(0,0)$
$\Rightarrow x+2=0$ and $y+3=0$
$\Rightarrow x=-2$ and $y=-3$
So, vertex $(-2,-3)$
Focus $=(a,0)$
$(x+2,y+3)=\left(\frac{1}{2},0\right)$
$\Rightarrow x+2=1/2$ and $y+3=0$
$\Rightarrow x=-\frac{3}{2},y=-3$
So, focus $\left(-\frac{3}{2},-3\right)$.
Equation of directrix,
$X=-a$
$\Rightarrow x+2=-\frac{1}{2}$
$\Rightarrow x=\frac{-1}{2}-2$
$\Rightarrow x=\frac{-5}{2}$
$\Rightarrow 2x+5=0$
Equation of axis,
$Y=0$
$\Rightarrow y+3=0$