Ellipse Voice Over

\( \text {For any P } \;\;F_1 P + F_2 P\) = const

\(F_1B +F_2B = F_1B +F_1A = AB =2a \)

\(F_1C + F_2C = 2a \;\;\;F_1C = F_2C = a \)

\(F_1O = \sqrt {{{F_1C}^2} -{OC}^2} = \sqrt{a^2 - b^2}\)

\( F_1 A = AO - F_1 O = a- {\sqrt{a^2 - b^2}} = r_1\)

\( F_1 B = F_1O + OB = a+ {\sqrt{a^2 - b^2}} = r_2 \)

\( r_2 - r_1 = 2{\sqrt{a^2 - b^2}} \; and \; r_2 + r_1 = 2a \)

\( r_1 \;\; \;and\; \;\; r_2 \;are\; min \;and \;max\; distances.\)

\( \; {Eccentricity} \; e = { {F_1O} \over AO } = {{\sqrt{a^2 - b^2}} \over a} = {{r_2 -r_1} \over {r_2 +r_1}} \)

\( \text {For any P } \;\;F_1 P + F_2 P\) = const

\(F_1B +F_2B = F_1B +F_1A = AB =2a \)

\(F_1C + F_2C = 2a \;\;\;F_1C = F_2C = a \)

\(F_1O = \sqrt {{{F_1C}^2} -{OC}^2} = \sqrt{a^2 - b^2}\)

\( F_1 A = AO - F_1 O = a- {\sqrt{a^2 - b^2}} = r_1\)

\( F_1 B = F_1O + OB = a+ {\sqrt{a^2 - b^2}} = r_2 \)

\( r_2 - r_1 = 2{\sqrt{a^2 - b^2}} \; and \; r_2 + r_1 = 2a \)

\( r_1 \; and \; r_2 \; are\; min \;and \;max\; distances.\)

\( \; Eccentricity \; e = { {F_1O} \over AO } = {{\sqrt{a^2 - b^2}} \over a} = {{r_2 -r_1} \over {r_2 +r_1}} \)

Ellipse is one of the conic sections produced when a plane cuts the cone at different angles to the base.

Circle - slice parallel to base.

Ellipse - slice through the curved surface at an angle to base.

Parabola - slice through curved surface and base.

Hyperbola - slice through curved surface and the base at right angles to the base. This is one branch of the hyperbola. For its pair, we need to imagine a double cone.