No Intersection between Line and Hyperbola

The line y = mx + c does not intersects the hyperbola \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 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), using the method of solving simultaneous equation 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( {{b^2} + {c^2}} \right) = 0\,\,\,{\text{ -  -  -  }}\left( {{\text{iii}}} \right) \\ \end{gathered}



no-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. If equation (iii) has imaginary roots, then the line (i) will not intersect the hyperbola (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( {{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( {{b^2} + {c^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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