Polar Conic Sections

Question

Consider the equation below.

r = 12/(3-sin(θ))

The eccentricity is e = 1/3 and It is an ellipse

Give an equation of the directrix (in Cartesian Coordinates)

Answer 1

The directrices are parallel to the minor axis, at a distance a/e from the center, where a is half the length of the major axis. So we just need to calculate a and find the center. There are two approaches I would use:

Method 1-Stay in polar coordinates

To find the major axis, we look for the largest distance between points on the ellipse on opposite sides of the focus at the origin (because the foci lie on the major axis). This happens when sin(theta) is maximized/minimized: that is, when it's +/- 1. sin(theta) = 1 when theta = pi/2 and -1 when theta = 3pi/2, so the polar points (12/(3-1) = 6,pi/2) and (12/(3+1) = 3,3pi/2) are the furthest apart. In cartesian coordinates, these are just P = (0,6) and Q = (0,-3), so the distance between them is 9, and a = 9/2. Now it should be easy to find the center (it's the midpoint of P and Q).

Method 2-Convert to Cartesian coordinates

Multiplying out, we get that

3r-rsin(theta) = 12, so

3r = rsin(theta)+12

squaring both sides,

9r^2 = (rsin(theta))^2+24rsin(theta)+144

now use that r^2 = x^2+y^2 and y = rsin(theta) to get

9x^2+9y^2 = y^2+24y+144,

which turns into

9x^2+8(y^2-3y) = 144

if we complete the square and divide by 144, we should get the standard equation for a (shifted so that center is not the origin) ellipse.

Answer 2

r = 12 / (3 - sinθ)

3r - rsinθ = 12

3√(x²+y²) - y = 12

9(x²+y²) = y² +24y + 144

9x² + 8y² - 24y = 144

9x² + 8(y² - 3y) = 144

9x² + 8(y² - 3y + 9/4) = 144 + 18

9(x - 0)² + 8(y - 3/2)² = 162

(x - 0)² / 18 + (y - 3/2)² / (81/4) = 1

Ellipse has center (0, 3/2) and vertical major axis with a² = 81/4, b² = 18.

Directrices are y = 3/2 ± a / e = 3/2 ± 3a. So, y = 3/2 ± 27/2. Therefore, y = 15 and y = -12.