# 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, depending on if the point lies outside, on or inside the circle.

The equation of a circle is

The equation of a 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 of the above equation, we get

This 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 the coefficients of $m$ will have real and distinct roots if the discriminant is positive. Using the discriminant formula 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 the coefficients of $m$ will have real and distinct roots if the discriminant is zero. Using the discriminant formula 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 the coefficients of $m$ will have real and distinct roots if the discriminant is negative. Using the discriminant formula 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.