No Intersection between Line and Circle

Consider the equation of a circle with center at origin \left( {0,0} \right) and radius r. Then the equation of this circle is written as

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

Now the equation of a line is represented by

y = mx + c\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)

To find the point of intersection of the line (ii) and the circle (i), using the method of solving simultaneous equations in which one equation is in quadratic and the other is in linear form, we take the value of y from equation (ii) and put this value into equation (i), i.e. the equation of a circle as follows:

\begin{gathered} {x^2} + {\left( {mx + c} \right)^2} = {r^2} \\ \Rightarrow {x^2} + {m^2}{x^2} + 2mcx + {c^2} = {r^2} \\ \Rightarrow \left( {1 + {m^2}} \right){x^2} + 2mcx + {c^2} - {r^2} = 0\,\,\,{\text{ - - - }}\left( {{\text{iii}}} \right) \\ \end{gathered}


no-intersection-line-circle

Since equation (iii) is a quadratic equation in x, we compare this equation with the standard quadratic equation a{x^2} + bx + c = 0 to obtain the coefficients of a,b and c. We use these values in the discriminant, i.e. {b^2} - 4ac.

\begin{gathered} {b^2} - 4ac = {\left( {2mc} \right)^2} - 4\left( {1 + {m^2}} \right)\left( {{c^2} - {r^2}} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4{m^2}{c^2} - 4\left( {{c^2} + {m^2}{c^2} - {r^2} - {m^2}{r^2}} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4{m^2}{c^2} - 4{c^2} - 4{m^2}{c^2} + 4{r^2} + 4{m^2}{r^2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - 4{c^2} + 4{r^2} + 4{m^2}{r^2} \\ \end{gathered}

If the discriminant is less than zero, i.e. {b^2} - 4ac < 0, then equation (iii) has imaginary roots and the line will not intersect the circle as shown in the given diagram.

Thus, the line does not intersect the circle if

\begin{gathered} - 4{c^2} + 4{r^2} + 4{m^2}{r^2} < 0 \\ \Rightarrow 4{r^2} + 4{m^2}{r^2} < 4{c^2} \\ \Rightarrow {c^2} > {r^2} + {m^2}{r^2} \\ \Rightarrow {c^2} > {r^2}\left( {1 + {m^2}} \right) \\ \end{gathered}

This is the required condition line if the line does not intersect the circle.