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.