# Two Tangent Lines to a Circle

Two tangents can be drawn to a circle ${x^2} + {y^2} = {r^2}$ from any point $P\left( {{x_1},{y_1}} \right)$. The tangents are real and distinct, coincident or imaginary according as the point lies outside, on or inside the circle.
We have the equation of circle is

The equation of tangent to the circle (i) is given as

If the tangent (ii) is drawn from the point $P\left( {{x_1},{y_1}} \right)$, then this point must satisfy the equation of tangent (ii), i.e.

Squaring both sides if the above equation, we get

Which is the quadratic equation in the variable $m$, so $m$ will have two values giving two tangents drawn from a point $P\left( {{x_1},{y_1}} \right)$.

Real and Distinct Tangents:
Comparing equation (iii) with coefficients of $m$, will have real and distinct roots if discriminant is positive, using discriminant formula and we get the following result

This shows that the point $P\left( {{x_1},{y_1}} \right)$ lies outside the circle (i). Thus, the tangents drawn will be real and distinct if the point lies outside the circle.

Real and Coincident Tangents:
Comparing equation (iii) with coefficients of $m$, will have real and distinct roots if discriminant is zero, using discriminant formula and we get the following result

This shows that the point $P\left( {{x_1},{y_1}} \right)$ lies outside the circle (i). Thus, the tangents drawn will be real and coincident if the point lies on the circle.

Imaginary Tangents:
Comparing equation (iii) with coefficients of $m$, will have real and distinct roots if discriminant is negative, using discriminant formula and we get the following result

This shows that the point $P\left( {{x_1},{y_1}} \right)$ lies outside the circle (i). Thus, the tangents drawn will be imaginary if the point lies inside the circle.