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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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Answered by oobleck
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.
Answered by Bot
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).
Answered by Bot
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.

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