Question

Which of the following represents the length of the subtangent of the hyperbola $S=\frac{x^2}{a^2}+\frac{y^2}{b^2}-1$ at the point $(x_1,y_1)$

• A. $\frac{b^2{x}_1}{a^2}$
• B. $\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}-1$
• C. $x_1-\frac{a^2}{x_1}$
• D. None of the above

Answer 1

Answer: (C)

$x_1-\frac{a^2}{x_1}$

For any standard Hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Let the tangent at point $P(x_1,y_1)$ meet the $x$-axis at $T(x,0)$,

$C(0,0)$ 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 $(x_1,0)$

The equation of tangent at $P(x_1,y_1)$ is $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1$.....$\left(1\right)$

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

For finding $x-$ ordinate, putting value of $y=0$ in eq. $\left(1\right)$ we get,

$\Rightarrow$ $x=\frac{a^2}{x_1}$

So point $T$ is $(\frac{a^2}{x_1},0)$

From the figure, we can know that the length of sub-tangent for a standard hyperbola is $TN$

Also from figure $TN=CN-CT$

As $C$ is $(0,0)$, $N$ is $(x_1,0)$ and $T$ is $(\frac{a^2}{x_1},0)$,

$\because$ All the points lies on $x$-axis, in a line.

$\therefore$ $CN=x_1$

$CT=\frac{a^2}{x_1}$

$\Rightarrow$ $TN=x_1-\frac{a^2}{x_1}$,

Hence the length of sub-tangent is $(x_1-\frac{a^2}{x_1})$, which is only written in option $C$, So the option $C$ is correct.