# Length of the Tangent to a Circle

Let the tangent drawn from the point $P\left( {{x_1},{y_1}} \right)$ meet the circle at the point $T$ as shown in the given diagram. The equation is given by

Consider the triangle $PTC$ formed in this way is a right triangle, so according to the given diagram we have

It is observed that $\left| {TC} \right|$ is the radius of the circle, so ${\left| {TC} \right|^2} = {g^2} + {f^2} - c$.

We also have

Putting all these values in (ii), we get

This gives the length of the tangent from the point $P\left( {{x_1},{y_1}} \right)$ to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$.

Similarly, we can show that the $PS$ is also of the same length.

Example: Find the length of the tangent from $\left( {12, - 9} \right)$ to the circle

Dividing the equation of the circle by 3, we get the standard form

The required length of the tangent from $\left( {12, - 9} \right)$ is