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

Question

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

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Answer 1 · 2018-03-31T14:47:22

$m=+-1$ 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 · 2018-03-31T14:48:23