Questions: The center of a circle is (-4,2) and the circle is tangent to the y-axis. Find the equation of the circle in the form (x-h)^2+(y-k)^2=r^2.

Solution Steps

To find the equation of the circle, we need to determine the radius. Since the circle is tangent to the \( y \)-axis, the distance from the center to the \( y \)-axis is the radius. The center is at \((-4, 2)\), so the radius is the absolute value of the \( x \)-coordinate, which is 4. The equation of the circle is then \((x + 4)^2 + (y - 2)^2 = 4^2\).

The center of the circle is given as \((-4, 2)\). Since the circle is tangent to the \(y\)-axis, the radius \(r\) is the distance from the center to the \(y\)-axis. This distance is the absolute value of the \(x\)-coordinate of the center, which is \(4\).

The general equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Substituting the values we have:

• \(h = -4\)
• \(k = 2\)
• \(r = 4\)

The equation becomes: \[ (x + 4)^2 + (y - 2)^2 = 4^2 \]

Final Answer