Focus of hyperbola is `(+-3,0)` and equation of tangent is `2x+y-4=0`, find the equation of hyperbola is

Options

A. `4x^(2)-5y^(2)=20`
B. `5x^(2)-4y^(2)=20`
C. `4x^(2)-5y^(2)=1`
D. `5x^(2)-4y^(2)=1 `

Answer

A

Explanation

Given , `(pm ae, 0)=(pm3, 0)`
`implies ae=3 `
` implies a^(2)e^(2)=9`
`implies b^(2)+a^(2)=9` .............(i)
`:. 2x+y-4=0`
`implies y=-2x+4 `
is the tangent to the hyperbola .
`:. (4)^(2)=a^(2)(-2)^(2)-b^(2)`
`implies 4a^(2)-b^(2)=16` ..............(ii)
On solving Eqs . (i) and (ii), we get
` a^(2)=5, b^(2)=4 `
`:.` Equation of hyperbola is ` (x^(2))/(5)-(y^(2))/(4)=1 `
`implies 4x^(2)-5y^(2)=20`