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Difficulty: MediumEquations of Circles in the Coordinate Plane

In the xyxy-plane, the graph of the equation x2+y2+8x−10y=wx^2 + y^2 + 8x - 10y = w, where ww is a constant, is a circle. If the circle is tangent to the xx-axis, what is the value of ww?

  1. -16Answer
  2. B
    -25
  3. C
    -36
  4. D
    66

Answer

-16
Completing the square on the given equation gives (x+4)2+(y−5)2=w+41(x+4)^2 + (y-5)^2 = w + 41. The center of this circle is (−4,5)(-4, 5). If a circle is tangent to the xx-axis, the radius is equal to the distance from the center to the xx-axis, which is the absolute value of the yy-coordinate of the center, ∣5∣=5|5| = 5. Thus, the radius squared is 52=255^2 = 25. Equating w+41w + 41 to 2525 gives w=−16w = -16.

Step-by-Step Solution

1
Complete the square for the xx and yy terms in the equation x2+y2+8x−10y=wx^2 + y^2 + 8x - 10y = w.
(x+4)2+(y−5)2=w+41(x+4)^2 + (y-5)^2 = w + 41
Completing the square allows us to find the center (h,k)(h, k) and the radius squared r2r^2 of the circle in the standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2.
2
Identify the center of the circle and determine the radius based on the tangency to the xx-axis.
Center is (−4,5)(-4, 5) and radius r=5r = 5.
The standard form shows the center is (−4,5)(-4, 5). Since the circle is tangent to the xx-axis, the radius is the vertical distance from the center to the line y=0y=0 (the xx-axis), which is ∣5∣=5|5| = 5.
3
Set r2=w+41r^2 = w + 41 and solve for ww.
w=−16w = -16
Since r=5r = 5, we have r2=25r^2 = 25. Equating w+41w + 41 to 2525 gives w=25−41=−16w = 25 - 41 = -16.

Key Concept

Equations of circles in the coordinate plane can be written in the standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2 by completing the square. When a circle is tangent to the xx-axis, its radius is equal to the absolute value of the yy-coordinate of its center (∣k∣|k|).
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Topics

Equations of Circles in the Coordinate Plane

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