Polar equation?

Question

Find a polar equatioin for the circle
Center: (5, pi/2)
Solution point: (0,0)

Please show your work.
And could you tell me what a solution point is!
Thank you

Answer 1

Solution point just means a point on the circle, I think. First figure out the radius of the circle. This is the same as the distance from the center to the one point you know on the circle, which is:

s = sqrt(25 + pi^2/4)

Now, in rectangular coordinates the circle is:

(x-5)^2 + (y-pi/2)^2 = s^2 = 25 + pi^2/4.

To write this in polar coordinates, use the formulas x = rcos(theta) and y = rsin(theta). So:

(rcos(theta) - 5)^2 + (rsin(theta) - pi/2)^2 = 25 + pi^2/4.

Answer 2

The center of the circle is (r,θ) = (5,π/2). A point on the circle is (0,0). The center lies on the y axis at (x,y) = (0,5).

Solution point means a point that satisifies the equation. In this case a solution point is a point on the circle.

(r,θ) = (5,π/2) is equivalent to (x,y) = (0,5).

The distance from the center to a point on the circle is:

d = | 0 - 5| = 5

So the radius is 5.

The Cartesian equation for the circle is:

x² + (y - 5)² = 25

Remember the identities:

x = rcosθ
y = rsinθ
r² = x² + y²

x² + (y - 5)² = 25
x² + y² - 10y + 25 = 25
x² + y² - 10y = 0

r² - 10rsinθ = 0
r - 10sinθ = 0
r = 10sinθ
_____________

If you can think in polar, all this work would not be necessary. For a circle thru the origin with the center on the y axis at the point (x,y) = (0,b) the equation of the circle is simply:

r = 2bsinθ

We could have used this and simply wrote down the equation. The center (x,y) = (0,5) so b = 5 and 2b = 10.

For a circle thru the origin with the center on the x axis at the point (x,y) = (a,0) the equation of the circle is simply:

r = 2acosθ

Answer 3

a million) Y = r sin x X = r cos x r^2 = X^2 + Y^2 r = 3(a million + 2 sin x) r + 2r sin x = 3 sq. the two aspects r^2 + 4 (r sin x)^2 + 4 r^2 sin x = 9 X^2 + 5Y^2 + 4 Y (X^2 + Y^2)^(a million/2) = 9 2) 2r - r cos x = 2 2 (X^2 + Y^2)^(a million/2) - X = 2