What is the condition for a line y=mx+c to be tangent to the hyperbola x2a2−y2b2=1.
c=±√b2−a2m2
A line y = mx + c becomes a tangent to a curve when on solving the equations of the line and curve we get one solution.
Here solving these equation gives,
x2a2−(mx+c)2b2=1
b2x2−a2(mx+c)2=a2b2
x2.(b2−a2m2)−2a2mcx−a2c2−a2b2=0
We get one solution if Δ=0.
i.e.,4a4m2c2+4(b2−a2m2)a2(c2−b2)=0
a2m2c2+(b2c2−64−a2m2c2+a2b2m2)=0
c2−b2+a2m2=0
∴ c2=b2−a2m2.
c=±√b2−a2m2