If `2x-y+1=0` is a tangent to hyperbola `(x^(2))/(a^(2))+(y^(2))/(16)=1`, then which of the following are sides of a right angled triangle ?

Options

A. `2a`, `4`, `1`
B. `2a`, `8`, `1`
C. `a`, `4`, `1`
D. `a`, `4`, `2`

Explanation

The equations of tangents of slope `m` to the given hyperbola are
`y=mx+-sqrt(a^(2)m^(2)-16)`
But, it is given that `2x-y+1=0` or `y=2x+1` is tangent to the hyperbla.
`:.m=2` and `sqrt(a^(2)m^(2)-16)=1`
`impliesm=2` and `4a^(2)-16=1`
`impliesm=2` and `a^(2)=(17)/(4)impliesm=2` and `a=(sqrt(17))/(2)`
Clearly, `2a,4,1` are the sides of a triangle as the sum of any two is greater than then third.