Find the equation of the hyperbola whose : focus `(a ,0)` , directrix is `2x-y+a=0\ ` and eccentricity`\ =4/3\ `

Explanation

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

And we know that,

Distance of point `P` from focus`=` Eccentricity `xx` Distance of point `P` from directrix.

`therefore sqrt{(x-a)^2+(y-0)^2}=4/3({2x-y+a}/sqrt5)`

`implies (x-a)^2+(y-0)^2=16/9({2x-y+a}/sqrt5)^2`

`implies x^2+a^2-2ax+y^2=16/45(4x^2+y^2-4xy+a^2+4ax-2ay)`

`implies 45x^2-90ax+45a^2+45y^2=64x^2+16y^2-64xy-64ax-32ay+16a^2=0`

`implies 19x^2-29y^2-64xy+154ax-32ay-29a^2=0`

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