Question

If two distinct tangents can be drawn from the point $(\alpha ,2)$ on different branches of the hyperbola
$\frac{x^2}{9}-\frac{y^2}{16}=1$, then

• A. $|\alpha |<\frac{3}{2}$
• B. $|\alpha |>\frac{2}{3}$
• C. $|\alpha |>3$
• D. $\alpha =1$

Answer 1

The correct option is A $|\alpha |<\frac{3}{2}$



For hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$ to have two distinct tangents on different branches the point should lie in the region $R_1$ and $R_3$
So from $(\alpha ,2)$ two tangents can be drawn if it lies in between $A$ and $B$ [where $A$ and $B$ are point of intersection asymptotes with $y=2$
Equation of asymptotes are $\frac{x^2}{9}-\frac{y^2}{16}=0$
$\Rightarrow (\frac{x}{3}-\frac{y}{4})(\frac{x}{3}+\frac{y}{4})=0$
$\Rightarrow 4x=\pm 3y$
solve both the asymptotes with $y=2,$ we get $x=\pm \frac{3}{2}$
$-\frac{3}{2}<\alpha <\frac{3}{2}$
$\Rightarrow |\alpha |<\frac{3}{2}$