No Intersection Between a Line and Ellipse

The line y = mx + c does not intersect the ellipse \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1, so the condition for no intersection is that {c^2} < {a^2}{m^2} + {b^2}.

Consider that the equation of a line is represented by

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

Consider that the standard equation of an ellipse with vertex at origin \left( {0,0} \right) can be written as

\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)

To find the point of the intersection of a straight line (i) and the given ellipse (ii), using the method of solving simultaneous equations we solve equation (i) and equation (ii). Putting the value of y from equation (i) in equation (ii), we have

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


no-intersection-line-ellipse

Since equation (iii) is a quadratic equation in x it can have at most two roots. This shows that the line (i) can intersect the ellipse (ii) at two points maximum. This is also clear from the given diagram. If equation (iii) has imaginary roots, then the line (i) will not intersect the ellipse (ii), as shown in the given diagram

For imaginary roots, we have

\begin{gathered} {\text{Discriminant < 0}} \\ \Rightarrow {\left( {2{a^2}mc} \right)^2} - 4\left( {{a^2}{m^2} + {b^2}} \right){a^2}\left( {{c^2} - {b^2}} \right) < 0 \\ \Rightarrow 4{a^4}{m^2}{c^2} - 4{a^2}\left( {{a^2}{m^2} + {b^2}} \right)\left( {{c^2} - {b^2}} \right) < 0 \\ \Rightarrow {a^2}{m^2}{c^2} - \left( {{a^2}{m^2} + {b^2}} \right)\left( {{c^2} - {b^2}} \right) < 0 \\ \Rightarrow - {b^2}{c^2} + {a^2}{m^2}{b^2} + {b^4} < 0 \\ \Rightarrow {c^2} - {a^2}{m^2} - {b^2} < 0 \\ \Rightarrow \boxed{{c^2} < {a^2}{m^2} + {b^2}} \\ \end{gathered}