Equation of tangent to a hyperbola in different forms

Hyperbola

x²/a² – y²/b² = 1

point (x1,y1)



Slope form

The line y = mx+c is a tangent to the hyperbola if c² = a²m²-b² .
Hence, the line y = mx±SQRT(a²m²-b²) is always a tangent.
The points of contact are (±a²m/c,±b²/c)

Point form

Tangent at (x1,y1) is

xx1/a²-yy1/b² = 1

Parametric form

point (a sec θ, b tan θ)

(x sec θ)/a) - ((y tan θ)/b) = 1