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Zorluk: KolayEquations and Graphs of Circles

A circle in the standard (x,y)(x, y) coordinate plane is defined by the equation (x−9)2+(y+4)2=81(x - 9)^2 + (y + 4)^2 = 81. What is the radius of this circle?

Cevap: 9

Cevap

The radius of the circle is 9.
The standard equation of a circle in the coordinate plane is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where rr is the radius. For the equation (x−9)2+(y+4)2=81(x - 9)^2 + (y + 4)^2 = 81, the right-hand side represents r2r^2, so r2=81r^2 = 81. Taking the positive square root of both sides gives r=81=9r = \sqrt{81} = 9.

Adım Adım Çözüm

1
Identify the standard form of the circle equation.
The standard equation of a circle is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center and rr is the radius.
This allows us to relate the terms of the given equation to the components of the circle.
2
Match the given equation to the standard form.
By comparing (x−9)2+(y+4)2=81(x - 9)^2 + (y + 4)^2 = 81 to (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, we find that r2=81r^2 = 81.
The constant on the right side of the standard equation represents the square of the radius.
3
Solve for the radius rr.
r=81=9r = \sqrt{81} = 9.
Taking the square root of r2r^2 gives the actual radius of the circle.

Anahtar Kavram

Identifying the radius from the standard equation of a circle
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Konular

Equations and Graphs of Circles

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