Question

Find the equation of the hyperbola whose focus is (2, −1), directrix is 2x + 3y = 1 and eccentricity = 2 .

Answer 1

Let S be the focus and  $$P(x,y)$$  be any point on the hyperbola. Draw PM perpendicular to the directrix.

By definition:
SP = ePM
= ePM

$$\Rightarrow$$ $$\sqrt{(x-2{)}^2+(y+1{)}^2}=2\left(\frac{2x+3y-1}{\sqrt{13}}\right)$$ 

Squaring both the sides:

$$(x-2{)}^2+(y+1{)}^2=4{\left(\frac{2x+3y-1}{13}\right)}^2$$

$$\Rightarrow x^2+4-4x+y^2+1+2y=\frac{4}{13}(4x^2+9y^2+1+12xy-6y-4x)$$

$$\Rightarrow 13x^2+52-52x+13y^2+13+26y=16x^2+36y^2+4+48xy-24y-16x$$

$$\Rightarrow 3x^2+23y^2+48xy-50y+36x-61=0$$

∴ Equation of the hyperbola = $$3x^2+23y^2+48xy-50y+36x-61=0$$