Question

For what values of `m` is line `y=mx` tangent to the hyperbola `x^2-y^2=1`?

Answer 1

`m=+-1` and tangents are `y=x` and `y=-x`

Explanation:

Put `y=mx` in te equation of hyperbola `x^2-y^2=1`, then

`x^2-m^2x^2=1`

or `x^2(1-m^2)-1=0`

the values of `x` will give points of intersection of `y=mx` and `x^2-y^2=1`. But as `y=mx` is a tangent, weshould get only one root, which would be wwhen discriminant is zero i.e.

`0^2-4*(1-m^2)*(-1)=0`

or `4-4m^2=0`

i.e. `m=+-1`

and tangents are `y=x` and `y=-x`

graph{(x^2-y^2-1)(x+y)(x-y)=0 [-10, 10, -5, 5]}

Answer 2

`"Slope of tangent " = dy/dx`

Therefore, slope of tangent of `x^2−y^2=1 => dx^2/dx−dy^2/dx=d1/dx`

`=> 2x-2y dy/dx =0`

`=> dy/dx =x/y`

For what values of m is line `y=mx` tangent to the hyperbola `x^2−y^2=1`?

`=> m =x/y`