A straight line l intersects the hyperbola $\frac {x^2}{a^2} -\frac { y^2}{b^2} = c$ at points P1 and P2.

Question

(i) Suppose M is the midpoint of P1 and P2, find the coordinates of M.

(ii) If the coordinates of M is (h,k), find the equation of l.

There are all unknown in this question. Should we let the equation l be $y=mx+c$ ? Or there are other way to solve?

Answer 1

With no information provided and if (h,k) is midpoint of the chord then locus of midpoint is $\frac{hx}{a^2} -\frac{ky}{b^2} =\frac{h^2}{a^2} -\frac{k^2}{b^2} $ The c term is eliminated

Answer 2

$\left\{ {\begin{array}{*{20}{l}} {\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = c} \\ {y = mx + n} \end{array}} \right.$

Substitute

$\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{(mx+n)^2}}}{{{b^2}}} = c$

Expand and collect

$x^2 \left(b^2-a^2 m^2\right)-2 a^2 m n x-a^2n^2-a^2 b^2 c=0$

The midpoint $M(h,\;k)$ has coordinates which are the mean, the average of the coordinates of the intersection point

$x_M=\dfrac{x_1+x_2}{2}$ remember that the sum of the roots $x_1+x_2=\dfrac{2a^2 m n}{b^2-a^2 m^2}$ so we have

$h=x_M=\dfrac{a^2 m n}{b^2-a^2 m^2}$

in a similar way we find

$k=y_M=\dfrac{b n}{b^2-a^2 m^2}$