Question

Find the coordinates of the centre, foci and equation of directrix of the hyperbola $x^2-3y^2-4x=8$.

Answer 1

$x^2-3y^2-4x-8=0$

$x^2-4x+4-4-3y^2-8=0$

${(x-2)}^2-3y^2=12$

$\frac{{(x-2)}^2}{\sqrt{1}{2}^2}-\frac{y^2}{2^2}=1$

Centre $(h,k)=(2,0)$

$a=2\sqrt{3}$

$b=2$

$c^2=b^2+a^2$

$c=\pm 4$

Foci $=(h,k+c)(h,k-c)$

Foci $=(2,4)(2,-4)$

Equation of directrix

$y=\frac{b^2}{{\sqrt{a}}^2+b^2}$

$y=\frac{-b^2}{{\sqrt{a}}^2+b^2}$

$y=1$

$y=-1$