# Condition for Line Tangent to Hyperbola

The condition for a line $y = mx + c$ to be the tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$is that $c = \pm \sqrt {{a^2}{m^2} - {b^2}}$ and the tangent to the hyperbola are $y = mx \pm \sqrt {{a^2}{m^2} - {b^2}}$.
Consider the equation of a line is represented by

Consider the standard equation of hyperbola with vertex at origin $\left( {0,0} \right)$can be written as

To find the point of intersection of straight line (i) and the given hyperbola (ii), using the method of solving simultaneous equation we solve equation (i) and equation (ii). Putting the value of $y$ from equation (i) in equation (ii), we have

If equation (iii) has equal roots, then the line equation (i) will intersect the hyperbola (ii) at one point only and so is the tangent to the hyperbola.
For equal roots, we have

Putting these values of $c$ in the equation of straight line (i), we have

These are the tangents to the hyperbola as shown in the given diagram.