Question

Find the length of sub -tangent of $\frac{x^2}{4}-\frac{y^2}{9}=1$ at $(4,0)$.

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is A $3$

For given Hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$

Let the tangent at point $P(x_1,y_1)$ meet the $x$-axis at $T$, $C(0,0)$ is the center and the perpendicular drawn from point $P$ to $x-$axis meets the $x-$ axis at $N$ .

The coordinates of point $N$ are $(x_1,0)$ and from the given information we know that sub-tangent is at $(4,0)$.

Hence $x_1=4$, So point $N$ is $(4,0)$

The equation of tangent at $P(x_1,y_1)$ is $\frac{x{x}_1}{4}-\frac{y{y}_1}{9}=1$.....$\left(1\right)$

Point $T$ lies on $x-$axis so its $y$ ordinate is $0$

For finding $x-$ ordinate, putting value of $y$ in eq. $\left(1\right)$ we get,

$\to$ $x=\frac{4}{x_1}=\frac{4}{4}=1$

So point $T$ is $(1,0)$

Here the length of sub-tangent is $TN$ and from figure $TN=CN-CT$

as $C$ is $(0,0)$, $N$ is $(4,0)$ and $T$ is $(1,0)$

Hence $TN=4-1=3$,

So option $A$ is correct.