Question

The number of tangents to the hyperbola $\frac{x^2}{4}-\frac{y^2}{3}=1$ through $(1,4)$ is

Answer 1

Position of point $(1,4)$ w.r.t. hyperbola
$\frac{x^2}{4}-\frac{y^2}{3}=1$ is
$S_1=\frac{1}{4}-\frac{16}{3}-1\Rightarrow S_1<0$
$\therefore$ The point $(1,4)$ lies outside the hyperbola
Hence the number of tangents that can be drawn from $(1,4)$ is $2.$