$$(\frac{x^2}{a^2} + \frac{y^2}{b^2} - 1)(\frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{r^2}{a^2b^2}) + \frac{r^2}{a^2b^2} = 0$$

$\begin{eqnarray*}x = \frac{x_1 + x_2}{2}\\ y = \frac{y_1 + y_2}{2}\end{eqnarray*}$

$\frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1$————(1)
$\frac{x_2^2}{a^2} + \frac{y_2^2}{b^2} = 1$————(2)

$(x_1 - x_2)^2 + (y_1 - y_2)^2 = 4r^2$————(3)

$\frac{x_1^2 + x_2^2}{a^2} + \frac{y_1^2 + y_2^2}{b^2} = 2$
$\frac{(x_1 + x_2)^2 + (x_1 - x_2)^2}{a^2} + \frac{(y_1 + y_2)^2 + (y_1 - y_2)^2}{b^2} = 4$————(4)

$\frac{4x^2 + 4w^2}{a^2} + \frac{4y^2 + 4h^2}{b^2} = 4$
$\frac{x^2 + w^2}{a^2} + \frac{y^2 + h^2}{b^2} = 1$————(5)

$\frac{x_1^2 - x_2^2}{a^2} + \frac{y_1^2 - y_2^2}{b^2} = 0$
$\frac{(x_1 + x_2)(x_1 - x_2)}{a^2} + \frac{(y_1 + y_2)(y_1 - y_2)}{b^2} = 0$
$\frac{2x\cdot2w}{a^2} + \frac{2y\cdot2h}{b^2} = 0$

$\frac{x^2}{a^2}\cdot\frac{w^2}{a^2} = \frac{y^2}{b^2}\cdot\frac{h^2}{b^2}$————(6)

$w^2 + h^2 = r^2$————(7)

$\begin{eqnarray*}\frac{x^2 + w^2}{a^2} + \frac{y^2 + h^2}{b^2} = 1......(8) \\ \frac{x^2}{a^2}\cdot\frac{w^2}{a^2} =\frac{y^2}{b^2}\cdot\frac{h^2}{b^2} ......(9)\\ w^2 + h^2 =r^2......(10) \end{eqnarray*}$

$\begin{eqnarray*}p = \frac{x}{a}......(11) \\q = \frac{y}{b}......(12) \\m = \frac{w}{a}......(13) \\n = \frac{h}{b}......(14)\end{eqnarray*}$

$\begin{eqnarray*}p^2 + q^2 + m^2 + n^2 = 1......(15) \\p^2m^2 = q^2n^2......(16) \\a^2m^2 + b^2n^2 = r^2......(17) \end{eqnarray*}$

$(17) \times p^2 + (17) \times q^2$得：
$a^2m^2p^2 + b^2n^2p^2 + a^2m^2q^2 + b^2n^2q^2 = p^2r^2 + q^2r^2$————(18)

$a^2n^2q^2 + b^2n^2p^2 + a^2m^2q^2 + b^2m^2p^2 = p^2r^2 + q^2r^2$
$(a^2q^2 + b^2p^2)(m^2 + n^2) = (p^2 + q^2)r^2$————(19)
$(15) \times (a^2q^2 + b^2p^2)$，得：
$(a^2q^2 + b^2p^2)(p^2 + q^2 + m^2 + n^2) = (a^2q^2 + b^2p^2)$
$(a^2q^2 + b^2p^2)(p^2 + q^2) + (a^2q^2 + b^2p^2)(m^2 + n^2) = (a^2q^2 + b^2p^2)$

$(a^2q^2 + b^2p^2)(p^2 + q^2) + (p^2 + q^2)r^2 = (a^2q^2 + b^2p^2)$
$(p^2 + q^2)(a^2q^2 + b^2p^2 + r^2) = (a^2q^2 + b^2p^2 + r^2) - r^2$
$(p^2 + q^2 - 1)(a^2q^2 + b^2p^2 + r^2) + r^2 = 0$
$(p^2 + q^2 - 1)(\frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{a^2b^2}) + \frac{r^2}{a^2b^2} = 0$

$p = \frac{x}{a}$
$q = \frac{y}{b}$

$$(\frac{x^2}{a^2} + \frac{y^2}{b^2} - 1)(\frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{r^2}{a^2b^2}) + \frac{r^2}{a^2b^2} = 0$$

PDF下载：椭圆定长弦中点轨迹的一种解法.pdf