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, `(+-ae,0)=(+-3,0)`
`impliesae=3`
`impliesa^(2)e^(2)=9`
`impliesb^(2)+a^(2)=9` . . . (i)
`because2x+y-4=0`
`impliesy=-2x+4`
is the tangent to the hyperbola.
`therefore(4)^(2)=a^(2)(-2)^(2)-b^(2)`
`implies4a^(2)-b^(2)=16` . . (ii)
On solving eqs. (i) and (ii), we get
`a^(2)=5,b^(2)=4`
`therefore` Equation of hyperbola is
`(x^(2))/(5)-(y^(2))/(4)=1`
`implies4x^(2)-5y^(2)=20`