Intersection of Line and Hyperbola

The line y = mx + c intersects the hyperbola \frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} =  1 at most two points and condition for such intersection is {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 hyperbola 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 hyperbola (ii), we have to solve equation (i) and equation (ii) by the method of simultaneously equations, and using the case when one equation is quadratic from and other one is in linear form, putting the value of y from equation (i) and putting this value in equation of hyperbola (ii) i.e. equation of hyperbola 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( {{b^2} - {c^2}} \right) = 0\,\,\,{\text{ -  -  -  }}\left( {{\text{iii}}} \right) \\ \end{gathered}



intersection-line-hyperbola

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 hyperbola (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( {{b^2} + {c^2}} \right) > 0 \\ \Rightarrow 4{a^4}{m^2}{c^2} - 4{a^2}\left(  {{a^2}{m^2} - {b^2}} \right)\left( {{b^2} + {c^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   - {a^2}{m^2} + {b^2} + {c^2} > 0 \\ \Rightarrow \boxed{{c^2} > {a^2}{m^2} -  {b^2}} \\ \end{gathered}

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