### Math Notes

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#### Analytic Geometry Solutions

##### Topics || Problems

In the asteroid belt between the orbits of Mars and Jupiter some asteroids are known to have orbits with eccentricities as high as 2/3. For such an elliptic orbit, find the ratio of the minor to major axis. Solution:

The eccentricity of an ellipse, $$e = \frac{c}{a}$$

The ratio of the minor to the major axis, $$\frac{2b}{2a}$$ = $$\frac{b}{a}$$

In an ellipse the relationship $$c^2 + b^2 = a^2$$ holds true.

$$(ea)^{2} + b^2 = a^2$$

$$b^2 = a^2 - (ea)^{2}$$

$$b^2 = a^2 (1 - e^2)$$

$$\frac{b}{a} = \sqrt{ (1 - e^2)}$$

$$\frac{b}{a} = 0.75$$