The condition for a line to be the tangent to the hyperbola is that and the tangent to the hyperbola are .
Consider the equation of a line is represented by
Consider the standard equation of hyperbola with vertex at origin 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 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 in the equation of straight line (i), we have
These are the tangents to the hyperbola as shown in the given diagram.