Question

The foci of the hyperbola are $S(-3,-2),S^1(5,6)$. If its $e=2$ 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 D $x+y-1=0$

Given foci $S(-3,-2),S^1(5,6)$

$2C=S{S}^1$

$=\sqrt{{(-3-5)}^2+{(-2-6)}^2}$

$=8\sqrt{2}$

$\therefore C=4\sqrt{2}$

Given $e=2$

We know that $e=\frac{C}{a}$

$\therefore a=2\sqrt{2}$

Distance between focus 'S' and directrix corresponding to it$=C-\frac{a}{e}$

$=3\sqrt{2}$

$S{S}^1=(y+2)=\frac{-2-6}{-3-5}(x+3)$

$y=x+1$

We know that directrix is perpendicular to $S{S}^1$

$\therefore$Slope of directrix$=-1$

Let A be point on axis of hyperbola through which directrix passes.

$AS=3\sqrt{2}$ and A lies on $y=x+1$

$\therefore$x-co ordinate of $A=(-3)+\frac{3\sqrt{2}}{\sqrt{1+{(-1)}^2}}$

$=0$

$\therefore y=0+1\Rightarrow \Rightarrow y=1$

$\therefore A=(0,1)\Rightarrow \therefore$ directrix$=(y-1)=-1(x-0)\Rightarrow y+x-1=0$

Equation of line with slope $(-1)$ and passing through A be $x+y-1=0$