Question

Find the polar equation for the parabola whose focus is the pole and the vertex is (3,π)(Hint: See Section 10.6 and remember x = rcos(θ) and y = rsin(θ) )

Answer 1

Since the focus is at the pole, the directrix is a horizontal line with equation y = -r. The distance from the vertex to the focus is equal to the distance from the vertex to the directrix, so we have:

3 = -r

Solving for r, we get:

r = -3

The vertex is at (3, π), so we have:

x = r cos(θ) = -3 cos(θ)
y = r sin(θ) = -3 sin(θ) + π

Squaring both sides of the equation for y and simplifying, we get:

y^2 - 2πy + 9 = -9x

Rearranging the terms, we get:

9x + y^2 - 2πy + 9 = 0

This is the polar equation for the parabola.