Write the equation in standard form of the circle which passes through (-2, 16) and has x-intercepts -2 and -32.

If it passes through (-2,16) and (-2,0) and (-32,0) then by symmetry the center must be at (-17,8)

Now we can use any of the points to find that r = 17 so

(x+17)^2 + (y-8)^2 = 17^2

see

http://www.wolframalpha.com/input/?i=plot+%28x%2B17%29^2+%2B+%28y-8%29^2+%3D+17^2,+x%3D-2,y%3D16,x%3D-32,y%3D0

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.

To find the equation of a circle in standard form, we need to know its center and its radius.

Step 1: Find the center of the circle.
Since the circle passes through (-2, 16), this point must be the center of the circle.

Step 2: Find the radius of the circle.
To find the radius, we can use the distance formula between the center (-2, 16) and one of the x-intercepts (-2 or -32).

Using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Distance = √[(-2 - (-2))^2 + (16 - 0)^2]
Distance = √[(0)^2 + (16)^2]
Distance = √[0 + 256]
Distance = √256
Distance = 16

So, the radius of the circle is 16.

Step 3: Write the equation of the circle.
The equation of a circle in standard form is given by:
(x - h)^2 + (y - k)^2 = r^2

Where (h, k) is the center of the circle and r is the radius.

Plugging in the values we have:
(x - (-2))^2 + (y - 16)^2 = 16^2
(x + 2)^2 + (y - 16)^2 = 256

Therefore, the equation in standard form of the circle is (x + 2)^2 + (y - 16)^2 = 256.

To write the equation of a circle in standard form, we need to determine the center and radius of the circle.

Given that the circle passes through (-2, 16), we can conclude that this point lies on the circumference of the circle.

To find the center of the circle, we take the average of the x-coordinates and the average of the y-coordinates of the points that lie on the circumference. So, the x-coordinate of the center is the average of -2 and -32, which is (-2 - 32) / 2 = -17.

Similarly, the y-coordinate of the center is the average of 16 (from the point on the circumference) and 0 (from the x-intercepts). So, the y-coordinate of the center is (16 + 0) / 2 = 8.

Therefore, the center of the circle is (-17, 8).

Next, we need to find the radius of the circle. The radius is the distance between the center and any point on the circumference. We can choose one of the given points on the circumference, such as (-2, 16).

To calculate the radius, we use the distance formula, which is the square root of the sum of the squares of the differences between the x-coordinates and y-coordinates of two points.

The distance between (-2, 16) and the center (-17, 8) is:

√[(-2 - (-17))^2 + (16 - 8)^2]
= √(15^2 + 8^2)
= √(225 + 64)
= √289
= 17

So, the radius of the circle is 17.

Now, we have the center (-17, 8) and the radius 17, which allows us to write the equation of the circle in standard form. Standard form is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) represents the coordinates of the center and r represents the radius.

Substituting the values we found, we have:

(x - (-17))^2 + (y - 8)^2 = 17^2
(x + 17)^2 + (y - 8)^2 = 289

Thus, the equation in standard form of the circle is (x + 17)^2 + (y - 8)^2 = 289.