Write the standard for of the equation of the circle that passes through the points at (0,8),(8,0),and (16,8). Then identify the center and radius of the circle.
By inspection, the centre of the circle is at (8,8), since the three given points form a right-triangle in which the hypothenus is the diameter (0,8),(16,8).
The radius is half the diameter, namely (16-0)/2=8
The standard form of a circle with centre at (xo,yo) and radius of R is
(x/xo)2 + (y/yo)2 = R2
Can you take it from here?
Note:
If you are looking for the general case of a circle passing through three distinct points P1(x1,y1), P2(x2,y2) and P3(x3,y3), it can be obtained by evaluating the following 4x4 determinant:
x2 + y2 x y 1
x12 + y1sup>2 x1 y1 1
x22 + y2sup>2 x2 y2 1
x32 + y3sup>2 x3 y3 1
Sorry, the determinant does not show up very well, and there was a typo:
x2 + y2 x y 1
x12 + y12 x1 y1 1
x22 + y22 x2 y2 1
x32 + y32 x3 y3 1
Please ignore the following format testing:
<pre>
x2+ y2 x1 y1 1
x12+y12 x1 y1 1
x22+y22 x2 y2 1
x32+y32 x3 y3 1
</pre>
To find the standard form of the equation of a circle, you can use the formula:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle, and r represents the radius.
To determine the center and radius of the circle, we need to find the values of h, k, and r.
Step 1: Find the center (h, k)
To find the center, we can use the midpoint formula between any two points on the circle. Let's choose (0, 8) and (8, 0):
h = (x1 + x2)/2
= (0 + 8)/2
= 8/2
= 4
k = (y1 + y2)/2
= (8 + 0)/2
= 8/2
= 4
The center of the circle is therefore (4, 4).
Step 2: Find the radius (r)
To find the radius, we can use the distance formula between the center and any of the given points. Let's use the distance between (0, 8) and the center (4, 4):
r = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(4 - 0)^2 + (4 - 8)^2]
= √[4^2 + (-4)^2]
= √[16 + 16]
= √32
= 4√2
The radius of the circle is 4√2.
Step 3: Write the standard form of the equation
Now that we have the values of h, k, and r, we can write the standard form of the equation:
(x - h)^2 + (y - k)^2 = r^2
(x - 4)^2 + (y - 4)^2 = (4√2)^2
(x - 4)^2 + (y - 4)^2 = 32
Therefore, the standard form of the equation of the circle passing through the points (0, 8), (8, 0), and (16, 8) is (x - 4)^2 + (y - 4)^2 = 32. The center of the circle is (4, 4), and the radius is 4√2.