Tangents from centre of hyperbola are asymptotes

An asymptote of a hyperbola is a tangent whose point of contact is at infinity and which passes through the centre. We show that the two tangents drawn from the centre are exactly the lines y = ±(b/a)x.

Setup

Take the hyperbola x²/a² − y²/b² = 1 with the centre at the origin O(0, 0). Consider a line through O: y = mx.

Tangent condition

A line y = mx + c touches the hyperbola if c² = a²m² − b². (Derivation: put y = mx + c in b²x² − a²y² = a²b² to get (b² − a²m²)x² − 2a²mc x − a²(c² + b²) = 0; equal roots give a²m²c² + (b² − a²m²)(c² + b²) = 0, i.e. c² = a²m² − b².)

For the line to pass through the centre, c = 0:

0 = a²m² − b² → m² = b²/a² → m = ± b/a

Conclusion

The tangents from the centre are y = (b/a)x and y = −(b/a)x, that is

x²/a² − y²/b² = 0

These are precisely the asymptotes of the hyperbola. (The point of contact is at infinity: with c = 0 and m² = b²/a², both the x² and x coefficients of the quadratic vanish, leaving −a²b² = 0, so both roots are infinite.) Hence the pair of tangents from the centre are its asymptotes.