Find the equation of the circle with endpoints of diameter (-1,3) and (7, -5)
center must be x,y, or (3,1)
diameter is more compliated:
diameter=sqrt[(-1-7)^2 + (3+5)^2 ]
radius is half that
(x-3)^2+(y-1)^2=radius^2 will work.
check that math, it is easy to err when typing
or
Once you have bobpursley's equation of
(x-3)^2 + (y+1)^2 = r^2
sub in one of the points on the circle, I will use (-1,3)
(-1-3)^2 + (3+1)^2 = r^2
16 + 16 = r^2 = 32
(x-3)^2 + (y-1)^2 = 32
I noticed that bob had the centre as (3,1),
should have been (3,-1)
To find the equation of a circle, we need the center and the radius. The center of the circle is located at the midpoint of the diameter, and the radius is half the length of the diameter.
First, let's find the center of the circle. To do this, we need to find the midpoint between the two given endpoints of the diameter.
Midpoint formula:
The midpoint between two points (x₁, y₁) and (x₂, y₂) is given by:
(midpoint_x, midpoint_y) = ((x₁ + x₂)/2, (y₁ + y₂)/2)
In this case, the two endpoints of the diameter are (-1,3) and (7,-5).
So, applying the midpoint formula, we have:
(midpoint_x, midpoint_y) = ((-1 + 7)/2, (3 + (-5))/2)
= (6/2, -2/2)
= (3, -1)
The center of the circle is then located at the point (3, -1).
Next, let's find the radius of the circle. The radius is half the length of the diameter, which can be calculated using the distance formula.
Distance formula:
The distance between two points (x₁, y₁) and (x₂, y₂) is given by:
distance = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
In this case, the two endpoints of the diameter are (-1,3) and (7,-5).
So, applying the distance formula, we have:
distance = sqrt((7 - (-1))² + (-5 - 3)²)
= sqrt(8² + (-8)²)
= sqrt(64 + 64)
= sqrt(128)
= 8√2
The radius of the circle is then 8√2 / 2 = 4√2.
Finally, we can write the equation of the circle using the formula:
(x - h)² + (y - k)² = r²
In this case, the center is (h, k) = (3, -1) and the radius is r = 4√2.
Therefore, the equation of the circle is:
(x - 3)² + (y + 1)² = (4√2)²
Simplifying the equation further:
(x - 3)² + (y + 1)² = 32