Question

The focus of parabola whose vertex is $(4,5)$ and whose equation of directrix $x+y+1=0$ is

• A. $(-9,10)$
• B. $(10,9)$
• C. $(9,10)$
• D. None of these

Answer 1

Answer: (C)

Equation of directrix $x+y+1=0$.
Since directrix and axis of parabola are at right angles.
$\therefore$ Equation of axis is $x-y+k=0$ passes through the vertex.

$\therefore k=1$ (using (4, 5) in $x-y+k=0)$
$\therefore$ Required equation of axis i s $x-y+1=0$
Now vertex is mid point of $NS$
$\therefore \frac{x-1}{2}=4$
$\Rightarrow x=9$ and $\frac{y}{2}=5$
$\Rightarrow y=10$
$\therefore$ Focus is $\left(9,10\right)$Equation of directrix $x+y+1=0$.
Since directrix and axis of parabola are at right angles.
$\therefore$ Equation of axis is $x-y+k=0$ passes through the vertex.

$\therefore k=1$ (using (4, 5) in $x-y+k=0)$
$\therefore$ Required equation of axis i s $x-y+1=0$
Now vertex is mid point of $NS$
$\therefore \frac{x-1}{2}=4$
$\Rightarrow x=9$ and $\frac{y}{2}=5$
$\Rightarrow y=10$
$\therefore$ Focus is $\left(9,10\right)$