Need help please:

(3,0) (7,6)
(x-7)2 + (y-6)2 = 7(2)
(x-7)2 + (y-6)2 = 49

Having difficulty with this problem.Can someone help?

geometry - Reiny, Saturday, March 31, 2012 at 8:07am
You don't say what the actual problem is.

I see the resemblance to finding the equation of a circle??

What are you trying to find?

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geometry - Kel, Saturday, March 31, 2012 at 6:59pm
It says to write an equation of a circle with diameter
__
AB

Is A at (3,0) and B at (7,6) ?

if so then the center is halfway between.
Xc = (3+7)/2 = 5
Yc = (0+6)/3 = 3
so of form
(x - 5)^2 + (y - 3)^2 = r^2
but r is half the length of AB
AB^2 = 4^2 + 6^2 = 16+36 = 52
AB = 7.21
(1/2) AB = r = 3.61
r^2 = 13
so in the end
(x - 5)^2 + (y - 3)^2 = 13

Thanks for your help!

To write an equation of a circle, you need the coordinates of its center and either its radius or diameter.

In this case, the given points are (3,0) and (7,6), which are the endpoints of the diameter AB.

To find the center of the circle, you can find the midpoints of AB using the midpoint formula.

The midpoint formula states that the coordinates of the midpoint would be the average of the x-coordinates and the average of the y-coordinates of the endpoints.

So, the x-coordinate of the center can be found by taking the average of the x-coordinates of (3,0) and (7,6):

x-coordinate of center = (3 + 7) / 2 = 5

Similarly, the y-coordinate of the center can be found by taking the average of the y-coordinates of (3,0) and (7,6):

y-coordinate of center = (0 + 6) / 2 = 3

Therefore, the center of the circle is (5,3).

Next, we need to find the radius or diameter of the circle.

The diameter AB can be found using the distance formula.

The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Let's calculate the distance AB:

distance AB = sqrt((7 - 3)^2 + (6 - 0)^2)
= sqrt(4^2 + 6^2)
= sqrt(16 + 36)
= sqrt(52)
= sqrt(4 * 13)
= 2 * sqrt(13)

Therefore, the radius of the circle is half the diameter, so the radius is:

radius = (2 * sqrt(13)) / 2 = sqrt(13)

Now that we have the center (5,3) and the radius sqrt(13), we can write the equation of the circle.

The standard form of a circle's equation is:

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

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

Plugging in the values, we get:

(x - 5)^2 + (y - 3)^2 = (sqrt(13))^2
(x - 5)^2 + (y - 3)^2 = 13

Therefore, the equation of the circle with diameter AB is (x - 5)^2 + (y - 3)^2 = 13.