# Two Tangent Lines to a Circle

Two tangents can be drawn to a circle from any point . 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 , 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 , so will have two values giving two tangents drawn from a point .

__Real and Distinct Tangents__**:**

Comparing equation (iii) with the coefficients of 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 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 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 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 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 lies outside the circle (i). Thus, the tangents drawn will be imaginary if the point lies inside the circle.