Write an equation for the circle that is centered at (−4, 5) and tangent to the x-axis.
since the center is at y=5, and it is tangent to the x-axis, the radius is 5.
Now, what is the standard equation for a circle?
To write the equation for a circle, we need to use the general equation for a circle, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle, and r represents the radius.
In this case, the circle is centered at (-4, 5) and is tangent to the x-axis. Since the x-axis is a horizontal line that has a y-coordinate of 0, we know that the distance from the center of the circle to the x-axis is equal to the radius.
To calculate the radius, we need to measure the vertical distance from the center of the circle to the x-axis. In this case, since the circle is tangent to the x-axis, this distance is just the absolute value of the y-coordinate of the center of the circle. In other words, the radius is |5| = 5.
Now that we know the center (-4, 5) and the radius 5, we can substitute these values into the general equation for a circle:
(x - (-4))^2 + (y - 5)^2 = 5^2
Simplifying this equation gives us the equation of the circle:
(x + 4)^2 + (y - 5)^2 = 25