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\begin{array}{ll} d_{2}-d_{1}=\sqrt{(x-(-c))^{2}+(y-0)^{2}}+\sqrt{(x-c)^{2}+(y-0)^{2}}=2 a & \text { Distance formula } \\ \sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=2 a & \text { Simplify expressions. } \\ \sqrt{(x+c)^{2}+y^{2}}=2 a+\sqrt{(x-c)^{2}+y^{2}} & \text { Move radical to opposite side. } \\ (x+c)^{2}+y^{2}=\left(2 a+\sqrt{(x-c)^{2}+y^{2}}\right)^{2} & \text { Square both sides. } \\ x^{2}+2 c x+c^{2}+y^{2}=4 a^{2}+4 a \sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}+y^{2} & \text { Expand the squares. } \\ x^{2}+2 c x+c^{2}+y^{2}=4 a^{2}+4 a \sqrt{(x-c)^{2}+y^{2}}+x^{2}-2 c x+c^{2}+y^{2} & \text { Expand remaining squares. } \\ 2 c x=4 a^{2}+4 a \sqrt{(x-c)^{2}+y^{2}}-2 c x & \text { Combine like terms. } \\ 4 c x-4 a^{2}=4 a \sqrt{(x-c)^{2}+y^{2}} & \text { Isolate the radical. } \\ c x-a^{2}=a \sqrt{(x-c)^{2}+y^{2}} & \text { Divide by 4. } \\ {\left(c x-a^{2}\right)^{2}=a^{2}\left(\sqrt{(x-c)^{2}+y^{2}}\right)^{2}} & \text { Square both sides. } \\ c^{2} x^{2}-2 a^{2} c x+a^{4}=a^{2}\left(x^{2}-2 c x+c^{2}+y^{2}\right) & \text { Expand the squares. } \\ c^{2} x^{2}-2 a^{2} c x+a^{4}=a^{2} x^{2}-2 a^{2} c x+a^{2} c^{2}+a^{2} y^{2} & \text { Distribute } a^{2} . \\ a^{4} + c^{2} x^{2} = a^{2} x^{2} + a^{2}c^{2} + a^{2} y^{2} & \text { Combine like terms. } \\ c^{2} x^{2} - a^{2} x^{2} - a^{2} y^{2} = a^{2} c^{2} - a^{4} & \text { Rearrange terms } \\ x^{2}\left(c^{2}-a^{2}\right)-a^{2} y^{2}=a^{2}\left(c^{2}-a^{2}\right) & \text { Factor common terms. } \\ x^{2} b^{2}-a^{2} y^{2}=a^{2} b^{2} & \text { set } b^{2} = c^2 - a^2 . \\ \frac{x^{2} b^{2}}{a^{2} b^{2}}-\frac{a^{2} y^{2}}{a^{2} b^{2}}=\frac{a^{2} b^{2}}{a^{2} b^{2}} & \text { Divide both sides by } a^2 b^2 \\ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \end{array}