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Eccentric Angle : Ellipse

If α and β ar...

Question

If $α$ and $β$ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is

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Solution

Equation of chord function $P (α)$ and $Q (β)$ is

$\frac{x}{a}$ cos $\frac{α + β}{2}$ + $\frac{y}{b}$ sin $\frac{α + β}{2}$ = cos $\frac{α - β}{2}$

S(ae,0) is on their line

e cos $\frac{α + β}{2} + 0$ = cos $\frac{α - β}{2}$

e = $\frac{c o s \frac{α - β}{2}}{c o s \frac{α + β}{2}}$ * $\frac{s i n \frac{α + β}{2}}{s i n \frac{α + β}{2}}$ * $\frac{2}{2}$ = $\frac{s i n α + s i n β}{s i n (α + β)}$

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α

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are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is

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are the eccentric angles of the extremities of a focal chord of an ellipse of eccentricity e then

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are the eccentric angles of the extremities of a focal chord of an ellipse, then eccentricity of the ellipse is

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are the eccentric angles of the extremities of a focal chord of the ellipse

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Q. The eccentric angles of extremities of a focal chord (other than Major axis) of an ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

θ_{1}

θ_{2}

.
If the eccentricity of the ellipse are

e_{1}

e_{2}

for the conditions

a > b

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