Answer:
A) r = 2/(1 - sinθ)
B) r = 16/(5 + 3 cosθ)
C) r = 2/(1 + 2 cosθ)
Step-by-step explanation:
A) We are given the parabola vertex as:
(r, θ) = (1, -π/2)
The vertex coordinates indicates that it's below the pole and thus the equation of the conic section is represented by;
r = ed/(1 - e sinθ)
Since the equation is a parabola, then the eccentricity is 1.
Thus,plugging in the relevant values;
1 = (1 × d)/(1 - (1 × sin-π/2))
1 = d/(1 - (-1))
1 = d/(1 + 1)
1 + 1 = d
d = 2
Therefore the general equation would be gotten by putting 2 for d and 1 for E.
Thus;
r = (1 × 2)/(1 - (1 × sinθ))
r = 2/(1 - sinθ)
B) vertices of Ellipse are given as;
(2,0) and (8, π)
The equation will be represented by;
r = ep/(1 + e cosθ)
Where e which is the eccentricity is given by: e = c/a
c = (8 - 2)/2
c = 6/2
c = 3
And, a = (8 + 2)/2
a = 10/2
a = 5
Thus,e = 3/5
Thus;
r = (3/5)p/(1 + (3/5) cosθ))
Multiply both numerator and denominator by 5;
r = 3p/(5 + 3 cosθ))
Now, using the vertex (2,0) and plugging it, we have;
2 = 3p/(5 + 3 cos0))
2 = 3p/(5 + 3(1))
2 = 3p/8
p = 16/3
Thus,the equation is;
r = 3(16/3)/(5 + 3 cosθ))
r = 16/(5 + 3 cosθ)
C) Since the directrix is x = p, then the function in the denominator would be cosine. This is because x = 4 which is greater than 0.
It means that there will be an addition sign in the denominator. We use the standard form of;
r = ep/(1 + e cosθ)
So, e = 2 and p = 1
Then;
r = (2 × 1)/(1 + 2 cosθ)
r = 2/(1 + 2 cosθ)
