7.5 Polar functions of quadratic curves

(Calculator) Find the eccentricities of the following quadratic curves. If the curve has no eccentricity, write “None”

(a) [math]x^2+y^2=12[/math]

(b) [math]x^2+2y-7=0[/math]

(c) [math]3x^2-4(y-1)^2=1[/math]

(d) [math](y+1)^2=4[/math]

(e) [math]x^2+2y^2-4y=-1[/math]

(Calculator) Earth revolves around the sun. The track of Earth is an ellipse, and the sun is one of the foci [math](0,0)[/math]. Suppose the eccentricity of the track of Earth is 0.0167 and the semi focal length is 2.5 million kilometers.

(a) Given that the second foci is on the positive [math]x[/math]-axis, find the equation of the track of Earth in parametric form. (round to the fourth decimal place)

(b) One astronomical unit is 149.5 million kilometers and there are 365 days in one year. How many days in a year is the distance between Earth and the sun less than 1 astronomical unit?

Rewrite the polar function [math]r=\dfrac{1}{2+2\sin\theta}[/math] in standard form.