Intersection of Line and Ellipse

The line y = mx + c intersects the ellipse \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} =  1 at most two points and condition for such intersection is that {c^2} > {a^2}{m^2} + {b^2}.
Consider the equation of a line is represented by

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


Consider the standard equation of 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 intersection of straight line (i) and the given ellipse (ii), using the method of solving simultaneous equation we solve equation (i) and equation (ii), in which one equation is in quadratic and other is in linear form, so take value of y from equation (i) and putting this value in equation of ellipse (ii) i.e. equation of ellipse becomes

\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}



intersection-line-ellipse

Since equation (iii) is a quadratic equation in x, and can have at most two roots. This shows that the line (i) can intersect the ellipse (ii) at most two points. It is also clear from the given diagram.
Equation (iii) will have two real roots if

\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}

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