Question

Find the equation of the hyperbola whose directrix is $2x+y=1$ focus $(1,2)$ and eccentricity $\sqrt{3}$

Answer 1

Let P (x, y) be any point on the hyperbola.

Draw PM
perpendicular from P on the directirix.

Then by
definition $SP=ePM\Rightarrow {\left(SP\right)}^2=e^2{\left(PM\right)}^2\Rightarrow {(x-1)}^2+{(y-2)}^2=3{\left\{\frac{2x+y-1}{\sqrt{4+1}}\right\}}^2$

$\Rightarrow 5(x^2+y^2-2x-4y+5)=3(4x^2+y^2+1+4xy-2y-4x)\Rightarrow 7x^2-2y^2+12xy-2x+14y-22=0$

Which is the required hyperbola.