The equation (x + 6)^2 + (y+4)^2 = 36 models the position and range of the source of a radio signal. Describe the position of the source and the range of the signals.
The center of the circle is at (-6,-4) which represents the position of the radio source. And the radius of the circle is 6 units, which represents the range of the radio signals.
The equation (x + 6)^2 + (y+4)^2 = 36 models the position and range of the source of a radio signal. Describe the position of the source and the range of the signals. step-by-step answer
Step 1: Recognize the equation as a standard form equation of a circle.
(x + h)^2 + (y + k)^2 = r^2
In this equation, the circle has a center at (-h, -k) and a radius of r.
Step 2: Compare the given equation with the standard form equation.
(x + 6)^2 + (y + 4)^2 = 36
We can see that h = -6, k = -4, and r = 6.
Step 3: Use the values of h and k to find the center of the circle.
The center of the circle is (-h, -k) = (-(-6), -(-4)) = (6, 4)
So, the position of the radio source is at (6, 4).
Step 4: Use the value of r to find the range of the radio signals.
The radius of the circle is 6 units, so the range of the signals is 6 units in all directions from the center.
Therefore, the range of the radio signals extends up to a distance of 6 units from the position of the radio source.
A diameter of a circle has endpoints P(-10 -2) and Q(4,6)
a. Find the center of the circle.
b. Find the radius. If your answer is not an integer express it in radical form.
c. Write an equation for the circle.
a. To find the center of the circle, we need to find the midpoint of the segment PQ using the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
where P(-10, -2) and Q(4, 6).
Midpoint = ((-10 + 4)/2, (-2 + 6)/2) = (-3,2)
Therefore, the center of the circle is at (-3,2).
b. To find the radius of the circle, we need to find the distance between the center and either P or Q. We can use the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
If we use the center (-3,2) and endpoint P(-10,-2), we get:
Distance = √[(-10 - (-3))^2 + (-2 - 2)^2] = √[49 + 16] = √65
Therefore, the radius of the circle is √65.
c. The equation of a circle with a center (h,k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Using the center we found in part a, and the radius we found in part b, we can write the equation of the circle:
(x - (-3))^2 + (y - 2)^2 = (√65)^2
Simplifying and solving for y:
(x + 3)^2 + (y - 2)^2 = 65
(x + 3)^2 = 65 - (y - 2)^2
x^2 + 6x + 9 = 65 - y^2 +4y - 4
x^2 + 6x + y^2 - 4y = 48
Therefore, the equation of the circle is x^2 + 6x + y^2 - 4y = 48.
To describe the position of the source and the range of the signals, we can analyze the given equation, which represents a circle in the Cartesian plane.
The general equation of a circle centered at the point (h, k) with radius r is:
(x - h)^2 + (y - k)^2 = r^2
Comparing this equation to the given equation, we can identify the following:
Center of the circle: (-6, -4) - obtained by equating (x + 6) and (y + 4) to zero.
Radius of the circle: 6 - obtained from the right side of the equation, which is equal to r^2.
Therefore, the position of the source of the radio signal is at the point (-6, -4), and the range of the signals extends to a distance of 6 units from the source.
To describe the position of the source and the range of the signals, we need to analyze the given equation, (x + 6)^2 + (y + 4)^2 = 36, which represents a circle in the Cartesian coordinate system.
In general, a circle with center (h, k) and radius r is represented by the equation (x - h)^2 + (y - k)^2 = r^2. Comparing this general equation with the given equation, we can deduce that the center of the circle is the point (-6, -4), and its radius is 6.
Therefore, based on the given equation, the position of the source is at the coordinates (-6, -4), which is the center of the circle. This indicates that the source is located at the point (-6, -4) in the Cartesian plane.
The range of the signals can be determined by looking at the radius of the circle. In this case, the radius is 6 units. This means that the maximum distance from the source to any point on the circumference of the circle is 6 units.
Hence, the range of the signals from the source is a circle with a radius of 6 units, centered at (-6, -4) in the Cartesian plane.