Question

The foci of the hyperbola are $S(5,6),S^{\prime }(-3,-2)$. If its eccentricity is $2$, then the equation of its directrix corresponding to focus $S$ is

• A. $x+y-3=0$
• B. $x+y-5=0$
• C. $x+y-7=0$
• D. $x+y-1=0$

Answer 1

The correct option is B $x+y-5=0$



$S{S}^{\prime }=\sqrt{(5+3{)}^2+(6+2{)}^2}=8\sqrt{2}$
$\Rightarrow S{S}^{\prime }=2ae=8\sqrt{2}$
$\Rightarrow a=2\sqrt{2}$ $(\because e=2)$
$\Rightarrow \frac{a}{e}=\sqrt{2}=OF$
$SF=4\sqrt{2}-\sqrt{2}=3\sqrt{2}$
Since, slope of axis $S{S}^{\prime }=\frac{-2-6}{-3-5}=1$
$\therefore$ slope of directrix $=-1$
Equation of directrix is given by $y=-x+c$
$\Rightarrow y+x-c=0$
Now, $3\sqrt{2}=\left|\frac{5+6-c}{\sqrt{2}}\right|$(perpendicular distance of a point from a line)
$\Rightarrow |11-c|=6$
$\Rightarrow 11-c=\pm 6$
$\Rightarrow c=5$ or $c=17$
Hence, Equation of Directrix can be $x+y-5=0$ or $x+y-17=0$
centre$\left(O\right)\equiv (1,2)$
we know that centre and focus lies on opposite side to the directrix. But for $x+y-17=0$, centre$\left(O\right)$ and focus$\left(S\right)$ lies on the same side.Therefore $x+y-17=0$ is rejected.
$\therefore$ Equation of directrix corresponding to focus $S$ is $x+y-5=0$