The equation (x + 9)^2 + (y - 4)^2 = 81 models the position and range of the source of a radio signal. Describe the position of the source and the range of the signals
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Oh, please! The original equation was
(x + 9)^2 + (y - 4)^2 = 81
and you dis a lot of useless (and erroneous) manipulation, and ended up with
(x + 9)^2 + (y - 4)^2 = 97
??? It started out as a circle, yet you could not seem to recognize that fact! The circle has radius 9.
You are correct, I apologize for that error in my previous response. The given equation is indeed the equation of a circle with center at (-9, 4) and radius 9.
To see this, we can rewrite the equation in standard form as:
(x + 9)^2 + (y - 4)^2 = 81
This is in the form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius. Comparing this to our equation, we see that the center is (-9, 4) and the radius is 9.
Therefore, the source of the radio signal is located at point (-9, 4) and the range of the signals extends to a distance of 9 units in all directions from the center.
To describe the position of the source, we need to find the coordinates of the center of the circle.
Step 1: Rewrite the equation in standard form.
(x + 9)^2 + (y - 4)^2 = 81
Step 2: Expand the equation.
(x^2 + 18x + 81) + (y^2 - 8y + 16) = 81
x^2 + 18x + y^2 - 8y + 97 = 81
Step 3: Rearrange the equation.
x^2 + 18x + y^2 - 8y + 16 = 0
Step 4: Identify the coefficients of x and y and divide by 2 to find the center coordinates.
The coefficient of x is 18, so the x-coordinate of the center is -18/2 = -9.
The coefficient of y is -8, so the y-coordinate of the center is -(-8)/2 = 4.
Therefore, the center of the circle is (-9, 4), which represents the position of the radio signal source.
To describe the range of the signals, we need to find the radius of the circle.
Step 5: Identify the constant term on the right side of the equation.
In this case, the right side of the equation is equal to 81.
Step 6: Take the square root of the constant term to find the radius.
The square root of 81 is 9.
Therefore, the radius of the circle is 9.
To summarize, the position of the source is represented by the coordinates (-9, 4) and the range of the signals is a circle with a radius of 9 units.
To describe the position and range of the source, we need to analyze the given equation: (x + 9)^2 + (y - 4)^2 = 81.
First, let's write the equation in standard form, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.
Comparing the given equation with the standard form equation, we can find the center (h, k) of the circle by equating the two expressions inside the parentheses:
x + 9 = 0 --> x = -9
y - 4 = 0 --> y = 4
Therefore, the center of the circle is at the point (-9, 4).
Next, we need to determine the radius of the circle. In the standard form, the radius is given by r^2. Thus, in this case, r^2 = 81.
To find the radius (r), we take the square root of both sides of the equation:
√r^2 = √81
r = ± 9
Hence, the radius of the circle is 9.
With this information, we can conclude that the source of the radio signal is located at the point (-9, 4), and the range of the signals extends up to a distance of 9 units in all directions from the center.
To describe the position of the source and the range of the signals, we need to rewrite the given equation in standard form:
(x + 9)^2 + (y - 4)^2 = 81
Expanding the square terms, we get:
x^2 + 18x + 81 + y^2 - 8y + 16 = 81
Simplifying, we get:
x^2 + y^2 + 18x - 8y + 16 = 0
Completing the square for x and y, we add and subtract the squares of half the x and y coefficients, respectively:
x^2 + 18x + 81 + y^2 - 8y + 16 = 0 + 81 + 16
(x + 9)^2 + (y - 4)^2 = 97
Comparing this to the standard form of a circle, (x - h)^2 + (y - k)^2 = r^2, we see that the center of the circle is (-9, 4) and the radius of the circle is sqrt(97), which represents the range of the radio signal. Therefore, the source of the radio signal is located at (-9, 4) and its range is sqrt(97).