Question

Prove that the portions of any line which are intercepted between the asymptotes and the curves are equal.

Answer 1

The portions of any line intercepted between the asymptotes and the curves are equal.

Answer 2

Let the equation of the hyperbola be $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

The equations of the asymptotes are $\frac{x}{a} - \frac{y}{b} = 0$ and $\frac{x}{a} + \frac{y}{b} = 0$. The combined equation of the asymptotes is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0$.

Consider a line $L$ given by the equation $y = mx + c$, where $m \neq \pm b/a$ (the line is not parallel to an asymptote) and $c \neq 0$ (the line does not pass through the center).

To find the points of intersection of the line $L$ with the hyperbola, we substitute $y = mx+c$ into the hyperbola equation: $\frac{x^2}{a^2} - \frac{(mx+c)^2}{b^2} = 1$

$b^2 x^2 - a^2 (m^2 x^2 + 2mcx + c^2) = a^2 b^2$

$(b^2 - a^2 m^2) x^2 - (2a^2 mc) x - (a^2 c^2 + a^2 b^2) = 0$

This is a quadratic equation in $x$. Let the roots be $x_{P1}$ and $x_{P2}$, which are the x-coordinates of the intersection points $P_1$ and $P_2$ on the hyperbola.

The sum of the roots is $x_{P1} + x_{P2} = \frac{2a^2 mc}{b^2 - a^2 m^2}$.

The y-coordinates are $y_{P1} = mx_{P1} + c$ and $y_{P2} = mx_{P2} + c$.

The midpoint $M_P$ of the segment $P_1P_2$ has coordinates:

$x_{M_P} = \frac{x_{P1} + x_{P2}}{2} = \frac{a^2 mc}{b^2 - a^2 m^2}$

$y_{M_P} = \frac{y_{P1} + y_{P2}}{2} = \frac{(mx_{P1} + c) + (mx_{P2} + c)}{2} = m\left(\frac{x_{P1} + x_{P2}}{2}\right) + c = m\left(\frac{a^2 mc}{b^2 - a^2 m^2}\right) + c = \frac{a^2 m^2 c + c(b^2 - a^2 m^2)}{b^2 - a^2 m^2} = \frac{b^2 c}{b^2 - a^2 m^2}$.

So $M_P = \left(\frac{a^2 mc}{b^2 - a^2 m^2}, \frac{b^2 c}{b^2 - a^2 m^2}\right)$.

Now, to find the points of intersection of the line $L$ with the asymptotes, we substitute $y = mx+c$ into the combined asymptote equation: $\frac{x^2}{a^2} - \frac{(mx+c)^2}{b^2} = 0$

$b^2 x^2 - a^2 (m^2 x^2 + 2mcx + c^2) = 0$

$(b^2 - a^2 m^2) x^2 - (2a^2 mc) x - a^2 c^2 = 0$

This is a quadratic equation in $x$. Let the roots be $x_{Q1}$ and $x_{Q2}$, which are the x-coordinates of the intersection points $Q_1$ and $Q_2$ on the asymptotes.

The sum of the roots is $x_{Q1} + x_{Q2} = \frac{2a^2 mc}{b^2 - a^2 m^2}$.

The y-coordinates are $y_{Q1} = mx_{Q1} + c$ and $y_{Q2} = mx_{Q2} + c$.

The midpoint $M_Q$ of the segment $Q_1Q_2$ has coordinates:

$x_{M_Q} = \frac{x_{Q1} + x_{Q2}}{2} = \frac{a^2 mc}{b^2 - a^2 m^2}$

$y_{M_Q} = \frac{y_{Q1} + y_{Q2}}{2} = \frac{(mx_{Q1} + c) + (mx_{Q2} + c)}{2} = m\left(\frac{x_{Q1} + x_{Q2}}{2}\right) + c = m\left(\frac{a^2 mc}{b^2 - a^2 m^2}\right) + c = \frac{b^2 c}{b^2 - a^2 m^2}$.

So $M_Q = \left(\frac{a^2 mc}{b^2 - a^2 m^2}, \frac{b^2 c}{b^2 - a^2 m^2}\right)$.

The midpoint of the segment intercepted by the hyperbola ($M_P$) is the same as the midpoint of the segment intercepted by the asymptotes ($M_Q$). Let this common midpoint be $M$.

The points $Q_1, P_1, P_2, Q_2$ lie on the line $L$. Since $M$ is the midpoint of $P_1P_2$, we have $MP_1 = MP_2$ and $\vec{MP_1} = -\vec{MP_2}$. Since $M$ is the midpoint of $Q_1Q_2$, we have $MQ_1 = MQ_2$ and $\vec{MQ_1} = -\vec{MQ_2}$.

Assuming the points are ordered $Q_1, P_1, P_2, Q_2$ along the line (which is the case when the line intersects the hyperbola between the asymptotes), the segment intercepted between the asymptote $Q_1$ and the curve $P_1$ is $Q_1P_1$. The segment intercepted between the curve $P_2$ and the asymptote $Q_2$ is $P_2Q_2$.

The vector $\vec{Q_1P_1} = \vec{MP_1} - \vec{MQ_1}$.

The vector $\vec{P_2Q_2} = \vec{MQ_2} - \vec{MP_2} = (-\vec{MQ_1}) - (-\vec{MP_1}) = \vec{MP_1} - \vec{MQ_1}$.

Thus, $\vec{Q_1P_1} = \vec{P_2Q_2}$.

This implies that the length of the segment $Q_1P_1$ is equal to the length of the segment $P_2Q_2$.

$Q_1P_1 = P_2Q_2$.

This proves that the portions of any line which are intercepted between the asymptotes and the curves are equal.

This proof is also valid for a vertical line $x=k$ with $|k|>a$. In this case, the midpoint of the intersection points with the hyperbola is $(k, 0)$ and the midpoint of the intersection points with the asymptotes is also $(k, 0)$. The segments intercepted are vertical and their lengths can be shown to be equal.