Translate each equation into polar coordinates.
Rearranging, and remembering that we can replace x2 + y2 with r2, we find
Since we have r2 and r terms, it's not reasonable to isolate r by itself.
If r ≠ 0 then we can divide both sides by r to find
It's reasonable to separate out r2 in this equation.
Replace y to find
Replace x and y to find
Replace x2 + y2 with r2 to find
r < 10.
r cos θ + r2 sin2 θ = 4r sin θ
r cos θ + (r sin θ)2 = 4r sin θ
x + y2 = 4y.
r sin θ = 5r cos θ + 2
y = 5x + 2
r = 4
r2 = 16
x2 + y2 = 16
Substitute in x and y before trying to rearrange.
r cos2 θ + r sin2 θ = 1
r(cos2 θ + sin2 θ) = 1
r = 1
Then square both sides:
r = 1
r2 = 1
x2 + y2 = 1
5r2 cos2 θ + 10r cos θ + 5 – 4r2sin2 θ + 16r sin θ – 16 = 20
5(r cos θ)2 + 10r cos θ + 5 – 4(r sin θ)2 + 16r sin θ – 16 = 20
5x2 + 10x + 5 – 4y2 + 16y – 16 = 20.
Now we can factor to tidy things.
5x2 + 10x + 5 – 4y2 + 16y – 16 = 20
5(x2 + 2x + 1) – 4(y2– 4y + 4) = 20
5(x + 1)2 – 4(y – 2)2 = 20.
Dividing both sides by 20, we see that this is the equation of a hyperbola:
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