উপবৃত্তের ফোকাসসমূহ

Given two points known as the foci, an ellipse is the set of all points in a plane containing the foci such that the sum of the distances from the point to the two foci is a constant.

More concisely, for two fixed points $F_1$ and $F_2$ and a constant $k\gt F_1F_2$, an ellipse is the locus of all points $P$ in a plane containing $F_1$ and $F_2$ such that $PF_1+PF_2=k$.

For the sake of simplicity, we will place the ellipse in coordinate space such that: $$F_1=(c,0)$$ $$F_2=(-c,0)$$ Let $P=(x,y)$. Then we have: $$\sqrt{(x-c)^2+y^2}+\sqrt{(x+c)^2+y^2}=k$$ Squaring both sides, $$(x-c)^2+y^2+(x+c)^2+y^2+2\sqrt{[(x-c)^2+y^2]((x+c)^2+y^2)}=k^2$$ Rearranging and simplifying, $$k^2-2x^2-2y^2-2c^2=2\sqrt{y^4+(2x^2+2c^2)y^2+(x+c)^2(x-c)^2}$$ Squaring both sides (let us all take a deep breath now), we have: $$k^4+4x^4+4y^4+4c^4-4k^2x^2-4k^2y^2-4k^2c^2+8x^2y^2+8x^2c^2+8y^2c^2\\=4y^4+(8x^2+8c^2)y^2+4(x+c)^2(x-c)^2$$ Now notice that a lot of stuff cancels out. Simplifying, $$k^4+4x^4+4c^4-4k^2x^2-4k^2y^2-4k^2c^2+8x^2c^2=4(x+c)^2(x-c)^2$$ Expanding the right side and simplifying even more, $$k^4+4x^4+4c^4-4k^2x^2-4k^2y^2-4k^2c^2+8x^2c^2=4x^4-8x^2c^2+4c^4$$ $$k^4-4k^2x^2-4k^2y^2-4k^2c^2+8x^2c^2=-8x^2c^2$$ Rearranging and simplifying one last time, we have: $$x^2(4k^2-16c^2)+y^2(4k^2)=k^4-4k^2c^2$$ $$\frac{4}{k^2}x^2+\frac{4}{k^2-4c^2}y^2=1$$ This is the most standard form for writing the equations of ellipses, with one on the right side, and a sum of some multiples of $x^2$ and $y^2$ on the left. In general, the equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ is an ellipse, with semi-axes of length $a$ and $b$. The longer axis is called the major axis, and the shorter one is called the minor axis. The foci are located at $$(\sqrt{a^2-b^2},0),(-\sqrt{a^2-b^2},0)$$ $$(0,\sqrt{b^2-a^2}),(0,-\sqrt{b^2-a^2})$$ depending on whether $a$ or $b$ is larger. The foci always lie on the major axis.