Line and the Hyperbola - Practice Questions & MCQ

Hyperbola : $\quad \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Line: $\quad y=m x+c$
After solving Eq. (i) and Eq. (ii)

$
\begin{aligned}
& \quad \frac{x^2}{a^2}-\frac{(m x+c)^2}{b^2}=1 \\
& \Rightarrow \quad\left(a^2 m^2-b^2\right) x^2+2 \mathrm{mca}^2 x+c^2 a^2+a^2 b^2=0
\end{aligned}
$

Above equation is quadratic in $x$
The line will cut the hyperbola in two points may be real, coincident or imaginary, that depends on the value of Discriminant, D.
1. If $D>0$, then two real and distinct roots which means two real and distinct points of intersection of the line and the hyperbola. In this case, the line is secant (chord) to the hyperbola.
2. If $\mathrm{D}=0$, then equal real roots which means the line is tangent to the hyperbola. Solving $\mathrm{D}=0$ we get the condition for tangency, which is $c^2=a^2 m^2-b^2$
3. If $D<0$, then no real root which means the line and hyperbola do not intersect.