A circle C has center at the origin and radius 9. Another circle K has a diameter with one end at the origin and the other end at the point (0,17). The circles C and K intersect in two points. Let P be the point of intersection of C and K which lies in the first quadrant. Let (r,theta) be the polar coordinates of P, chosen so that r is positive and 0<theta<2 inclusive. Find r and theta.
WEll, r has to be 9(center with radius 9)
x^2+y^2=9
x^2+(y-17/2)^2= (289/4)
subtract the first equation from the second
(Y-17/2)^2-y^2=289/4 - 9
-17y+289/4=289/4-9
y= 9/17 at the point of intersection.
sinTheta= 9/17*3= 9/51
check my work.
To find the polar coordinates (r, θ) of the point of intersection P, we can use the Cartesian coordinates of P.
First, let's find the coordinates of P. We have one point at the origin (0, 0) and another point at (0, 17). To find the point of intersection, we need to find the distance between the origin and (0, 17) and then divide it in half to find the midpoint.
The distance between the origin and (0, 17) is the length of the diameter of circle K, which is simply the y-coordinate of the point (0, 17).
Diameter of K = 17
To find the midpoint, we divide the diameter by 2:
Midpoint = 17/2 = 8.5
So the y-coordinate of the point P is 8.5.
Now, let's find the x-coordinate of P. Since the center of circle C is at the origin, the x-coordinate of P will be the same as the radius of circle C (which is 9).
So the coordinates of P are (9, 8.5).
To find the polar coordinates (r, θ), we can use the following formulas:
r = √(x^2 + y^2)
θ = tan^(-1)(y/x) + π/2 (to account for the first quadrant)
Plugging in the values:
r = √((9)^2 + (8.5)^2) = √(81 + 72.25) = √(153.25) ≈ 12.4 (rounded to one decimal place)
θ = tan^(-1)(8.5/9) + π/2 ≈ 0.768 (rounded to three decimal places)
Therefore, the polar coordinates of the point of intersection P are approximately (r, θ) = (12.4, 0.768).
To find the polar coordinates (r, theta) of the point P, we need to determine the values of r and theta.
Given that Circle C has its center at the origin (0, 0) and radius 9, we can express any point on Circle C in polar coordinates as (r, theta), where r represents the distance from the origin to the point and theta represents the angle between the positive x-axis and the line joining the origin to the point.
Since Point P lies on Circle C, its polar coordinates can be expressed as (r, theta).
We are also given that Circle K has a diameter with one end at the origin and the other end at the point (0, 17). This means the diameter of Circle K is the distance between the origin and (0, 17), which is 17.
Now, let's proceed to find the values of r and theta for Point P.
Since Point P lies on both Circle C and Circle K, its distance from the origin should be equal to the radius of Circle C and half the diameter of Circle K. Therefore:
r = 9 = 17/2
Simplifying the equation, we have:
2r = 17
r = 17/2 = 8.5
Now, to determine the value of theta, we need to find the angle between the positive x-axis and the line joining the origin to Point P.
Since Point P lies in the first quadrant, the angle theta will be positive and less than 90 degrees.
To calculate the value of theta, we can use the trigonometric function tangent:
tan(theta) = opposite / adjacent
The opposite side is the y-coordinate of Point P, which is 17, and the adjacent side is the x-coordinate of Point P, which is unknown.
Using the equation tan(theta) = 17/x, we can solve for x:
x = 17 / tan(theta)
Now, we have the value of x. To calculate theta, we can use the arctangent function:
theta = arctan(17/x)
Substituting the value of x, we have:
theta = arctan(17 / (17 / tan(theta)))
Simplifying further:
theta = arctan(tan(theta))
Since theta lies in the first quadrant, we can use the equation:
theta = theta
Therefore, the value of theta for Point P is simply:
theta = theta
Combining the values of r and theta, we have:
r = 8.5 and 0 < theta < 2
Hence, the polar coordinates of Point P, chosen so that r is positive and 0 < theta < 2, are (8.5, theta).