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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+3)2+(y−4)2=25(x + 3)^2 + (y - 4)^2 = 25. What are the coordinates of the center and the length of the radius of this circle?

  1. A
    Center: (3,−4)(3, -4); Radius: 55
  2. B
    Center: (−3,4)(-3, 4); Radius: 2525
  3. Center: (−3,4)(-3, 4); Radius: 55Answer
  4. D
    Center: (3,−4)(3, -4); Radius: 2525
  5. E
    Center: (−3,−4)(-3, -4); Radius: 55

Answer

Center: (−3,4)(-3, 4); Radius: 55
The standard form of a circle's equation is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where the center is (h,k)(h, k) and the radius is rr. Matching (x+3)2+(y−4)2=25(x + 3)^2 + (y - 4)^2 = 25 to this standard form, we find h=−3h = -3, k=4k = 4, and r2=25r^2 = 25 (which gives r=5r = 5). Therefore, the center is (−3,4)(-3, 4) and the radius is 55.

Step-by-Step Solution

1
Identify the standard form of a circle's equation.
The standard equation of a circle with center (h,k)(h, k) and radius rr is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.
This formula provides the template to match the given equation and extract the parameters.
2
Rewrite the given equation (x+3)2+(y−4)2=25(x + 3)^2 + (y - 4)^2 = 25 to match the signs and exponents of the standard form.
The equation can be written as (x−(−3))2+(y−4)2=52(x - (-3))^2 + (y - 4)^2 = 5^2.
Rewriting the terms helps to identify the exact values of hh, kk, and rr without sign confusion.
3
Extract the center (h,k)(h, k) and radius rr from the rewritten equation.
Comparing the terms shows h=−3h = -3, k=4k = 4, and r=5r = 5, giving a center of (−3,4)(-3, 4) and a radius of 55.
These extracted values are the final answer.

Key Concept

Equations and Graphs of Circles
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Topics

Equations and Graphs of Circles

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