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Difficulty: HardEquations and Graphs of Circles

In the standard (x,y)(x, y) coordinate plane, the equation x2+y2+8x−12y+c=0x^2 + y^2 + 8x - 12y + c = 0 defines a circle. If this circle is tangent to the yy-axis, what is the value of the constant cc?

  1. A
    1616
  2. 3636Answer
  3. C
    4848
  4. D
    5252
  5. E
    6868

Answer

The value of the constant cc is 3636.
By completing the square on x2+y2+8x−12y+c=0x^2 + y^2 + 8x - 12y + c = 0, we get the standard form equation (x+4)2+(y−6)2=52−c(x+4)^2 + (y-6)^2 = 52-c. This shows the circle's center is at (−4,6)(-4, 6) and its radius squared is r2=52−cr^2 = 52-c. Because the circle is tangent to the yy-axis, the radius is the horizontal distance from the center to the yy-axis, which is 44 units. Therefore, the radius squared is 1616. Equating the two expressions for the radius squared gives 52−c=1652-c = 16, which solves to c=36c = 36.

Step-by-Step Solution

1
Group the xx and yy terms and move the constant to the right side of the equation.
(x2+8x)+(y2−12y)=−c(x^2 + 8x) + (y^2 - 12y) = -c
To prepare the equation for completing the square.
2
Complete the square for both the xx and yy groups by adding the square of half the coefficient of the linear terms to both sides.
(x2+8x+16)+(y2−12y+36)=−c+16+36(x^2 + 8x + 16) + (y^2 - 12y + 36) = -c + 16 + 36, which simplifies to (x+4)2+(y−6)2=52−c(x+4)^2 + (y-6)^2 = 52-c.
To rewrite the equation in standard circle form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2.
3
Determine the radius of the circle based on the geometric condition of being tangent to the yy-axis.
The center of the circle is (−4,6)(-4, 6). Since the circle is tangent to the yy-axis (the line x=0x=0), the radius rr is the horizontal distance from the center to the yy-axis, which is ∣−4∣=4|-4| = 4.
A circle tangent to a vertical line has its radius equal to the horizontal distance from its center to that line.
4
Set the radius squared equal to the right side of the standard equation and solve for cc.
r2=42=16r^2 = 4^2 = 16. Setting 52−c=1652 - c = 16 gives c=36c = 36.
In standard form, the right-hand side represents r2r^2.

Key Concept

Converting the general form of a circle equation to standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2 by completing the square, and using the geometric definition of tangency to determine the radius.
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Topics

Equations and Graphs of Circles

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