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Difficulty: EasyEquations and Graphs of Circles

A circle in the standard (x,y)(x,y) coordinate plane is defined by the equation (x−8)2+(y+5)2=121(x - 8)^2 + (y + 5)^2 = 121. What is the diameter of this circle?

Answer: 22

Answer

The diameter of the circle is 22.
By comparing the given equation (x−8)2+(y+5)2=121(x - 8)^2 + (y + 5)^2 = 121 to the standard circle equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, we find that r2=121r^2 = 121. Taking the square root of 121121 gives a radius of r=11r = 11. Since the diameter of a circle is twice the radius, the diameter is 2×11=222 \times 11 = 22.

Step-by-Step Solution

1
Identify the relationship between the circle's equation and its radius squared.
r2=121r^2 = 121
The standard equation of a circle is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, meaning the constant on the right side of the equation represents the square of the radius.
2
Determine the radius of the circle.
r=11r = 11
Taking the square root of 121121 yields the radius of the circle, which must be a positive length.
3
Calculate the diameter of the circle.
d=22d = 22
The diameter of a circle is defined as twice the length of its radius (d=2rd = 2r).

Key Concept

The standard equation of a circle is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where rr is the radius and the diameter is 2r2r.
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Topics

Equations and Graphs of Circles

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