No Intersection between Line and Circle

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

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


Now equation of a line is represented by

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


To find the point of intersection of line (ii) and the circle (i), using the method of solving simultaneous equation in which one equation is in quadratic and other is in linear form, so take value of y from equation (ii) and putting this value in equation (i) i.e. equation of 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, and compare this equation with the standard quadratic equation a{x^2} + bx + c = 0, to obtained the coefficients of a,b and c. Using 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) will have imaginary roots and the line will not intersect the circle as shown in the given diagram. Thus, 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}


Which is the required condition line is not intersects the circle.

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