A circle touches the x-axis at point \( (4,0) \) and its center lies on the line \( x + y = 6 \). What is the equation of the circle?
Show Hint
- \((x - 4)^2 + (y - 3)^2 = 9\)
- \((x - 4)^2 + (y - 2)^2 = 4\)
- \((x - 4)^2 + (y - 2)^2 = 9\)
- \((x - 3)^2 + (y - 3)^2 = 9\)
The Correct Option is B
Solution and Explanation
Step 1: Let the center of the circle be \((h, k)\) lying on the line \(x + y = 6\), so \[ h + k = 6. \] Step 2: Since the circle touches the x-axis at point \((4,0)\), the radius \(r\) is the perpendicular distance from the center to the x-axis. The x-axis is the line \(y=0\), so \[ r = |k - 0| = |k|. \] Step 3: The point \((4,0)\) lies on the circle, so \[ (4 - h)^2 + (0 - k)^2 = r^2 = k^2. \] Step 4: Substituting radius \(r^2 = k^2\), we get \[ (4 - h)^2 + k^2 = k^2 \implies (4 - h)^2 = 0 \implies h = 4. \] Step 5: From step 1, \[ h + k = 6 \implies 4 + k = 6 \implies k = 2. \] Step 6: Radius \(r = |k| = 2\), so radius squared \[ r^2 = 4. \] Step 7: The equation of the circle is \[ (x - 4)^2 + (y - 2)^2 = 4. \] But this corresponds to option (A), not (B). Let's verify carefully. Step 3 says: \[ (4 - h)^2 + (0 - k)^2 = r^2, \] and \(r = |k|\), so \(r^2 = k^2\). Therefore, \[ (4 - h)^2 + k^2 = k^2 \implies (4 - h)^2 = 0 \implies h = 4. \] From \(h + k = 6\), \[ k = 2. \] Thus radius \(r = 2\), and radius squared \(= 4\). So the equation is indeed \((x - 4)^2 + (y - 2)^2 = 4\), matching option (B). Hence, the correct answer is (B).
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