For the general ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ,show that the midpoints of the chords lie on a straight line. 

Question: A  collection  of  parallel    chords  connect pairs   of  points  on  an  ellipse,    as  shown

For  the general ellipse  $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,    show    that    the midpoints   of  the chords  lie on  a   straight    line.



What I have tried/attempted:
Say that a line in the form $y=mx+c$ intersects the ellipse
so 
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
$$ \Leftrightarrow x^2b^2 + y^2a^2 = a^2b^2 $$
$$ \Leftrightarrow x^2b^2 + (mx+c)^2a^2 = a^2b^2 $$
$$\Leftrightarrow x^2b^2 + (m^2x^2+2mcx+c^2)a^2=a^2b^2 $$
$$ \Leftrightarrow x^2b^2 + a^2m^2x^2+2mca^2x+c^2a^2 - a^2b^2 = 0 $$
$$ \Leftrightarrow (b^2+a^2m^2)x^2 + (2mca^2)x + (c^2a^2-a^2b^2) = 0 $$
$$\Leftrightarrow (b^2+a^2m^2)x^2 + (2mca^2)x + a^2(c^2-b^2) = 0 $$
Now I am stuck should I be doing something with this equation I have formed? Or is there a more easy geometrical approach to this question I am missing.

$$\Leftrightarrow (b^2+a^2m^2)x^2 + (2mca^2)x + a^2(c^2-b^2) = 0 $$
By the midpoint of a parabola, the $x$-value of the midpoint of the intersection of the line and the ellipse will be at
$$\frac{-2mca^2}{2(b^2+a^2m^2)} = \frac{-mca^2}{b^2+a^2m^2}$$
Note that the $y$ value of the midpoint is linearly dependent on the $x$ value of the midpoint $(y=mx+c)$. Therefore, this expression shows that for any fixed values of $a$, $b$, and $m$, the collection of midpoints for ranging $c$ values is linearly dependent on the variable $c$.
For a concrete example, take the ellipse $\frac{x^2}{4} +\frac{y^2}{9}=1$ intersected by lines with slope $m=3$. The $x$ coordinate for the midpoint of the chords will be
$$x_c=\frac{-mca^2}{b^2+a^2m^2}=\frac{-3\cdot c\cdot4}{9+4\cdot 9}=\frac{-4c}{15}$$
and we can find the corresponding $y$ coordinates of the midpoint by plugging it into our linear equation
$$y_c=mx+c=3\cdot \frac{-4c}{15}+c=\frac{1}{5}c$$
So we have the collection of midpoints $(\frac{-4c}{15},\frac{1}{5}c)$ which is obvious to show lies on a straight line for changing values of $c$.
