For the parabola $y^2+6 y-2 x+5=0$, match the items in List-I with the suitable item in List-II given below:…
For the parabola $y^2+6 y-2 x+5=0$, match the items in List-I with the suitable item in List-II given below:
The correct matching is
I $\quad$ II $\quad$ III $\quad$ IV
F A E C
F A C E
A B C D
F A C D
Solution
Parabola, $y^2+6 y-2 x+5=0$
$
\begin{array}{ll}
\Rightarrow & y^2+6 y=2 x-5 \\
\Rightarrow & y^2+6 y+9=2 x-5+9
\end{array}
$
$
\begin{array}{ll}
\Rightarrow & (y+3)^2=2(x+2) \\
\Rightarrow & Y^2=2 X
\end{array}
$
(where, $Y=y+3$ and $X=x+2$ )
This parabola is in the form of $y^2=4 a x$ by comparing
$
\begin{aligned}
& 4 a=2 \\
& a=\frac{1}{2} \\
& \text { Vertex }=(0,0) \\
& \Rightarrow \quad x+2=0 \text { and } y+3=0 \\
& \Rightarrow \quad x=-2 \text { and } y=-3 \\
& \text { So, vertex }(-2,-3) \\
& \text { Focus }=(a, 0) \\
& (x+2, y+3)=\left(\frac{1}{2}, 0\right) \\
& \Rightarrow \quad x+2=1 / 2 \text { and } y+3=0 \\
& \Rightarrow \quad x=-\frac{3}{2}, y=-3 \\
&
\end{aligned}
$
So, focus $\left(-\frac{3}{2},-3\right)$.
Equation of directrix,
$
\begin{array}{cc}
& X=-a \\
\Rightarrow & x+2=-\frac{1}{2} \Rightarrow x=\frac{-1}{2}-2 \\
\Rightarrow & x=\frac{-5}{2} \\
\Rightarrow & 2 x+5=0
\end{array}
$
Equation of axis,
$
Y=0 \Rightarrow y+3=0
$
