https://youtu.be/VN2ZghEUZlY Polar Equations of Conics Section 9.8

More uses for conics… Using our previous definitions of conics, we saw that the equations for conics take a simple form when the origin is at the center There are other important applications for conics for which it is convenient to use one of the foci as the origin, so that it is at the pole.

Alternative Definition of a Conic The locus of a point in the plane which moves so that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a conic. The constant ratio is the eccentricity of the conic (e).

e < 1  the conic is an ellipse e = 1  the conic is a parabola https://www.youtube.com/watch?v=ua79GIKjqtE If… e < 1  the conic is an ellipse e = 1  the conic is a parabola e > 1  the conic is a hyperbola (P) point on the directrix (Q) point on the conic (P) (Q) (F) focus (F) (F) (P) directrix (P) (Q) (Q) because PF < PQ… PF < 1 PQ because PF = PQ… PF = 1 PQ because PF > PQ… PF > 1 PQ

When the focus is at the pole… …the equation of the conic takes on a simpler form. POLAR EQUATIONS OF CONICS: r = ep OR r = ep 1 ± e cos θ 1 ± e sin θ where e > 0 is the eccentricity and |p| is the distance between the focus (pole) and the directrix

Analyzing this equation…

r = ep r = ep 1 ± e cos θ 1 ± e sin θ Vertical Directrix Horizontal Directrix r = ep 1+ e cos θ r = ep 1─ e cos θ r = ep 1+ e sin θ r = ep 1 ─ e sin θ directrix to the right of the pole directrix to the left of the pole directrix above the pole directrix below pole

Example #1 Identify the type of conic represented by the equation r = 20 + 4 cos θ Hint! 5 Answer: To rewrite in standard form, divide the numerator and denominator by 5 (to make the red five a one). r = 4 1 + (4/5) cos θ Because e = 4/5 < 1, the conic is an ellipse.

Example #2 Analyze the graph of this same equation r = 4 1 + (4/5) cos θ Answer: We already know that this conic is an ellipse. Because it has cosine in the equation, its directrix is vertical, and because there is a + in the denominator, the directrix is to the right of the pole. Also, because ep = 4 and e=4/5, we know that p=5. The directrix is 5 units away from the pole.

Major and Minor Axes The length of the major axis still = 2a, so find the distance between the two coordinates and divide by 2. 2a = 5+1, a = 3 (1, π) (5, 0)

For an ellipse, to find the length of the minor axis, use the formula b² = a²(1 ─ e²) can be derived from b² = a² ─ c² For a hyperbola, use b² = a²(e² ─ 1)