Question

a curve, described by x^2+y^2+24x=0, has a point P at (-12,12) on the curve. Part A: what are the polar coordinates of P? give an exact answer. Part B: What is the polar form of the equation? What type of polar curve is this? Part C: What is the distance when theta= pi/3?

Answer 1

Explanation:

Part A: To convert from rectangular coordinates to polar coordinates, we use the formulas:

r^2 = x^2 + y^2

tan(theta) = y/x

We can rewrite the equation of the curve as:

x^2 + 24x + y^2 = 0

Completing the square in x, we get:

(x + 12)^2 + y^2 = 144

So, r^2 = 144 and r = 12. To find theta, we use the formula:

tan(theta) = y/x

tan(theta) = 12/(-12)

theta = atan(-1) + pi

So, the polar coordinates of P are (12, 5pi/4).

Part B: To convert the equation to polar form, we use the formulas:

x = r cos(theta)

y = r sin(theta)

Substituting these into the equation of the curve, we get:

r^2 + 24r cos(theta) = 0

r = -24 cos(theta)

This is the polar form of the equation. This is a circle centered at (-12, 0) with radius 12.

Part C: To find the distance when theta = pi/3, we substitute this value into the polar equation:

r = -24 cos(pi/3)

r = -12

So, the distance is 12.

Answer 2

= pi/3 is 12.

Explanation:

Part A: To find the polar coordinates of the point P, we can use the conversion formulas:

x = r cos(theta)

y = r sin(theta)

We can rewrite the equation of the curve as:

x^2 + y^2 + 24x = 0

(x + 12)^2 + y^2 = 144

This is the equation of a circle with center (-12, 0) and radius 12. The point P is on this circle, with coordinates (-12, 12). To convert these Cartesian coordinates to polar coordinates, we can use:

r = sqrt(x^2 + y^2)

theta = arctan(y/x)

Substituting the coordinates of P, we get:

r = sqrt((-12)^2 + 12^2) = 12 sqrt(2)

theta = arctan(12/-12) = -pi/4 (since P is in the second quadrant)

Therefore, the polar coordinates of P are (r, theta) = (12 sqrt(2), -pi/4).

Part B: To find the polar form of the equation, we can first complete the square to rewrite it as:

x^2 + y^2 + 24x = 0

(x + 12)^2 - 144 + y^2 = 0

(x + 12)^2 + y^2 = 144

This is the equation of a circle with center (-12, 0) and radius 12, in Cartesian form. To convert this to polar form, we can use the conversion formulas:

x = r cos(theta)

y = r sin(theta)

Substituting and simplifying, we get:

(r cos(theta) + 12)^2 + (r sin(theta))^2 = 144

r^2 + 24r cos(theta) = 0

r = -24 cos(theta)

Therefore, the polar form of the equation is r = -24 cos(theta). This is the equation of a cardioid, a type of polar curve with a cusp at the origin.

Part C: To find the distance from the origin when theta = pi/3, we can substitute this value into the polar form:

r = -24 cos(pi/3) = -12

However, distance is always non-negative, so we take the absolute value to get:

| r | = 12

Therefore, the distance from the origin when theta = pi/3 is 12.

(Refrence to Chatgpts work)