Find the equation of tangent to the hyperbola `y=(x+9)/(x+5) ` which passes through `(0, 0)` origin

Options

A. `x+25y=0`
B. `x+y=0`
C. `5x-y=0`
D. `x-25y=0`

Answer

A, B

Explanation

`y=(x+9)/(x+5)=1+(4)/(x+5)`
`therefore" "(dy)/(dx)=(-4)/((x+5)^(2))`
At `(x_(1),y_(1)),(dy)/(dx)=(-4)/((x_(1)+5)^(2))`
Equation of tangent at `(x_(1),y_(1))`,
`y-y_(1)=(-4)/((x_(1)+5)^(2))(x-x_(1))`
`rArr" "y-1-(4)/(x_(1)+5)=(-4)/((x_(1)+5)^(2))(x-x_(1))`
Since it passes through (0, 0),
`(x_(1)+5)^(2)+4(x_(1)+5)+4x_(1)=0`
`rArr" "x_(1)^(2)+18x_(1)+45=0`
`therefore" "x_(1)=-15or x_(1)=-3`
So, equations of required tangents are `x+25y=0 or x+y=0`.