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


length-tangent-to-circle

\[{x^2} + {y^2} + 2gx + 2fy + c = 0\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]

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

\[{\left| {PT} \right|^2} + {\left| {TC} \right|^2} = {\left| {PC} \right|^2}\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)\]

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 \[{\left| {PC} \right|^2} = {\left( {{x_1} – \left( { – g} \right)} \right)^2} + {\left( {{y_1} – \left( { – f} \right)} \right)^2} = {\left( {{x_1} + g} \right)^2} + {\left( {{y_1} + f} \right)^2}\]

Putting all these values in (ii), we get

\[\begin{gathered} {\left| {PT} \right|^2} + {g^2} + {f^2} – c = {\left( {{x_1} + g} \right)^2} + {\left( {{y_1} + f} \right)^2} \\ \Rightarrow {\left| {PT} \right|^2} + {g^2} + {f^2} – c = {x_1}^2 + 2g{x_1} + {g^2} + {y_1}^2 + 2f{y_1} + {f^2} \\ \Rightarrow {\left| {PT} \right|^2} = {x_1}^2 + {y_1}^2 + 2g{x_1} + 2f{y_1} + c \\ \Rightarrow \left| {PT} \right| = \sqrt {{x_1}^2 + {y_1}^2 + 2g{x_1} + 2f{y_1} + c} \\ \end{gathered} \]

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
\[3{x^2} + 3{y^2} – 7x + 22y + 9 = 0\]

Dividing the equation of the circle by 3, we get the standard form
\[{x^2} + {y^2} – \frac{7}{3}x + \frac{{22}}{3}y + 3 = 0\]

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

\[\sqrt {{{\left( {12} \right)}^2} + {{\left( { – 9} \right)}^2} – \frac{7}{3}\left( {12} \right) + \frac{{22}}{3}\left( { – 9} \right) + 3} = \sqrt {144 + 81 – 28 – 66 + 3} = \sqrt {134} \]