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Difficulty: HardEquations of Circles in the Coordinate Plane

In the xyxy-plane, the graph of the equation 3x2+3y2−24x+18y−c=03x^2 + 3y^2 - 24x + 18y - c = 0, where cc is a constant, represents a circle. If the area of the circle is 100π100\pi, what is the value of cc?

  1. A
    75
  2. 225Answer
  3. C
    300
  4. D
    375

Answer

225
The correct answer is 225. Dividing the given equation 3x2+3y2−24x+18y−c=03x^2 + 3y^2 - 24x + 18y - c = 0 by 3 gives x2+y2−8x+6y−c3=0x^2 + y^2 - 8x + 6y - \frac{c}{3} = 0. Completing the square for the xx-terms by adding (−8/2)2=16(-8/2)^2 = 16 and for the yy-terms by adding (6/2)2=9(6/2)^2 = 9 to both sides results in (x−4)2+(y+3)2=c3+25(x - 4)^2 + (y + 3)^2 = \frac{c}{3} + 25. 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. Since the area of the circle is 100π100\pi and the area formula is A=πr2A = \pi r^2, the radius squared r2r^2 must equal 100. Setting the right-hand side of our standard form equation equal to 100 gives c3+25=100\frac{c}{3} + 25 = 100. Solving for cc yields c3=75\frac{c}{3} = 75, which gives c=225c = 225.

Step-by-Step Solution

1
Divide the entire equation by the coefficient of the squared terms to normalize it.
x2+y2−8x+6y−c3=0x^2 + y^2 - 8x + 6y - \frac{c}{3} = 0
The standard form of a circle's equation requires the coefficients of x2x^2 and y2y^2 to be 1.
2
Complete the square for both the xx and yy terms.
(x2−8x+16)+(y2+6y+9)=c3+16+9(x^2 - 8x + 16) + (y^2 + 6y + 9) = \frac{c}{3} + 16 + 9, which simplifies to (x−4)2+(y+3)2=c3+25(x - 4)^2 + (y + 3)^2 = \frac{c}{3} + 25
Completing the square converts the general equation of a circle into the standard form (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.
3
Use the given area of the circle to determine the radius squared, r2r^2.
r2=100r^2 = 100
The area of a circle is defined by A=πr2A = \pi r^2. Given that the area is 100π100\pi, we have πr2=100π\pi r^2 = 100\pi, which means r2=100r^2 = 100.
4
Equate the expression for r2r^2 from the standard form to the value found from the area and solve for cc.
c3+25=100⇒c3=75⇒c=225\frac{c}{3} + 25 = 100 \Rightarrow \frac{c}{3} = 75 \Rightarrow c = 225
From the standard form, the right-hand side is equal to r2r^2. Equating the two expressions allows us to solve for the constant cc.

Key Concept

To find the radius or related constants of a circle from its general equation, first divide by any common coefficient of the squared terms, then complete the square to write the equation in the standard form (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.
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Topics

Equations of Circles in the Coordinate Plane

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