No Intersection between a Line and Hyperbola

The line $y = mx + c$ does not intersect 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 a 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 equations we solve equation (i) and equation (ii). Putting the value of $y$ from equation (i) in equation (ii), we have

$\begin{gathered} \frac{{{x^2}}}{{{a^2}}} – \frac{{{{\left( {mx + c} \right)}^2}}}{{{b^2}}} = 1 \\ \Rightarrow \frac{{{b^2}{x^2} – {a^2}{{\left( {mx + c} \right)}^2}}}{{{a^2}{b^2}}} = 1 \\ \Rightarrow {b^2}{x^2} – {a^2}{\left( {mx + c} \right)^2} = {a^2}{b^2} \\ \Rightarrow {b^2}{x^2} – {a^2}\left( {{m^2}{x^2} + 2mcx + {c^2}} \right) = {a^2}{b^2} \\ \Rightarrow {b^2}{x^2} – {a^2}{m^2}{x^2} – 2{a^2}mcx – {a^2}{c^2} – {a^2}{b^2} = 0 \\ \Rightarrow \left( {{a^2}{m^2} – {b^2}} \right){x^2} + 2{a^2}mcx + {a^2}\left( {{b^2} + {c^2}} \right) = 0\,\,\,{\text{ – – – }}\left( {{\text{iii}}} \right) \\ \end{gathered}$ 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 hyperbola (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 hyperbola (ii) as shown in the given diagram.

For imaginary roots, we have

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