Why did the circle go to the party? Because it wanted to pass through all the x-intercepts and mingle with the (-2, 16) point! Let's find the equation in standard form for this circle.
The standard form equation for a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r is the radius.
To find the center of the circle, we can average the two x-intercepts: (-2 + -32)/2 = -34/2 = -17.
So, the x-coordinate of the center is -17. Since the circle passes through (-2, 16), we can use this point to find the y-coordinate of the center.
Plugging in (-2, 16) into the equation gives us (-2 + 17)^2 + (16 - k)^2 = r^2.
Simplifying this, we get (15)^2 + (16 - k)^2 = r^2.
To find the radius, we can use either one of the x-intercepts. Let's use -2.
Plugging in (-2, 0) into the equation, we get (-2 + 17)^2 + (0 - k)^2 = r^2.
Simplifying this, we get (15)^2 + (-k)^2 = r^2.
Now, we have two equations:
(15)^2 + (16 - k)^2 = r^2,
(15)^2 + (-k)^2 = r^2.
Canceling out the r^2 terms, we get:
(16 - k)^2 = (-k)^2.
Expanding this equation, we get:
256 - 32k + k^2 = k^2.
Simplifying, we find 256 - 32k = 0.
Thus, k = 8.
Now, we can substitute this value of k back into either equation to find the radius.
Using the equation (15)^2 + (16 - k)^2 = r^2:
(15)^2 + (16 - 8)^2 = r^2,
225 + 8^2 = r^2,
225 + 64 = r^2,
289 = r^2.
Therefore, the radius is √289 = 17.
The equation in standard form for the circle passing through (-2, 16) and having x-intercepts -2 and -32 is:
(x + 17)^2 + (y - 8)^2 = 289.