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Find the equation of the hyperbola if focus : $(2 , 3 );$ corresponding directrix : $x+2y=5,e=2$

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  • General equation of a conic with focus at $F(x,y)$,directrix $lx+my+n=0$ and eccentricity $e$ is $(x-x_1)^2+(y-y_1)^2=e^2\bigg[\pm\large\frac{lx+my+n}{\sqrt{l^2+m^2}}\bigg]^2$
  • Which reduces to the form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$
Step 1:
$F(2,3)$ and directrix $l : x+2y=5,e=2$.
Let $P$ be a point on the hyperbola and $PM\perp l$
Step 2:
The equation of the hyperbola is $x^2-16xy-11y^2+20x+50y-35=0$
answered Jun 18, 2013 by sreemathi.v

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