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

{x^2} + {y^2} = {r^2}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

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

y = mx + r\sqrt {1 + {m^2}} \,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)

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.:

\begin{gathered} {y_1} = m{x_1} + r\sqrt {1 + {m^2}} \\ \Rightarrow {y_1} - m{x_1} = r\sqrt {1 + {m^2}} \\ \end{gathered}

Squaring both sides of the above equation, we get

\begin{gathered} {\left( {{y_1} - m{x_1}} \right)^2} = {r^2}\left( {1 + {m^2}} \right) \\ \Rightarrow {y_1}^2 - 2m{x_1}{y_1} + {x_1}^2{m^2} = {r^2} + {r^2}{m^2} \\ \Rightarrow {x_1}^2{m^2} - {r^2}{m^2} - 2m{x_1}{y_1} + {y_1}^2 - {r^2} = 0 \\ \Rightarrow \left( {{x_1}^2 + {r^2}} \right){m^2} - 2m{x_1}{y_1} + {y_1}^2 - {r^2} = 0\,\,\,\,{\text{ - - - }}\left( {{\text{iii}}} \right) \\ \end{gathered}

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:

{x_1}^2 + {y_1}^2 > {r^2}

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:

{x_1}^2 + {y_1}^2 = {r^2}

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:

{x_1}^2 + {y_1}^2 < {r^2}

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.