The condition for a line to be the tangent to the ellipse is that and the tangent to the ellipse is .
Consider the equation of a line is represented by
Consider that the standard equation of an ellipse with vertex at origin can be written as
To find the point of intersection of a straight line (i) and the given ellipse (ii), using the method of solving simultaneous equations 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 ellipse (ii) at one point only and thus is the tangent to the ellipse.
For equal roots, we have
Putting these values of in the equation of a straight line (i), we have
These are the tangents to the ellipse as shown in the given diagram.