find the equation of the circle having the line segment joining the points (-2,2) and (-8,6)as diameter.
center halfway between
average x = -10/2 = -5
average y =8/2= 4
so center at (-5 , 4)
form therefore is
(x+5)^2 + (y-4)^2 = r^2
put point in to find r
(-2+5)^2 + (2-4)^2 = r^2
9 + 4 =r^2
r ^2 = 13
so
(x+5)^2 + (y-4)^2 = 13
========================
check, other point, y = 6
(x+5)^2 + 4 = 13
(x+5)^2 = 9
X+5 = 3 or -3
-3 ok, x = -8 as advertized
Why did the circle bring a line segment to the party as its diameter? Because it wanted to show off its geometric style! Now let's get down to business and find the equation of the circle.
To find the equation of a circle, we need to determine its center and radius. We can start by finding the midpoint of the line segment joining the two points (-2,2) and (-8,6). The midpoint formula is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Let's plug in the given points:
Midpoint = ((-2 + -8)/2, (2 + 6)/2)
= (-10/2, 8/2)
= (-5, 4)
So, the center of the circle is (-5, 4).
Now, let's find the radius of the circle, which is half the length of the line segment. The formula for distance between two points is:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the given points (-2,2) and (-8,6), we have:
Distance = sqrt((-8 - (-2))^2 + (6 - 2)^2)
= sqrt((-6)^2 + 4^2)
= sqrt(36 + 16)
= sqrt(52)
So, the radius of the circle is sqrt(52) or 2*sqrt(13).
Now we can write the equation of the circle using the standard form:
(x - h)^2 + (y - k)^2 = r^2
Substituting the center and radius values, we get:
(x - (-5))^2 + (y - 4)^2 = (2*sqrt(13))^2
(x + 5)^2 + (y - 4)^2 = 52
There you go! The equation of the circle is (x + 5)^2 + (y - 4)^2 = 52. Now that's a circular equation if I've ever seen one!
To find the equation of a circle, we need the center and the radius.
Step 1: Find the midpoint of the line segment joining the given points (-2,2) and (-8,6).
The midpoint formula is:
midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Using the coordinates (-2,2) and (-8,6), we can calculate the midpoint as follows:
x-coordinate of midpoint = (-2 + -8)/2 = -10/2 = -5
y-coordinate of midpoint = (2 + 6)/2 = 8/2 = 4
So, the midpoint is (-5, 4).
Step 2: Find the distance between the midpoint and one of the given points. This will give us the radius of the circle.
The distance formula is:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates (-5, 4) and (-2, 2), we can calculate the distance as follows:
distance = √((-2 - -5)^2 + (2 - 4)^2)
= √((3)^2 + (-2)^2)
= √(9 + 4)
= √(13)
So, the radius of the circle is √13.
Step 3: Finally, we can write the equation of the circle using the center and the radius in the standard form:
(x - h)^2 + (y - k)^2 = r^2
Here, the center is (-5, 4) and the radius is √13. Substituting these values, we get:
(x - (-5))^2 + (y - 4)^2 = (√13)^2
Simplifying, we have:
(x + 5)^2 + (y - 4)^2 = 13
Therefore, the equation of the circle having the line segment joining the points (-2,2) and (-8,6) as diameter is (x + 5)^2 + (y - 4)^2 = 13.
To find the equation of a circle, we need the center coordinates and the radius. Since we have been given the line segment joining the points (-2,2) and (-8,6) as the diameter, we can find the center coordinates by finding the midpoint of the line segment.
Let's start by finding the midpoint of the line segment:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Given points:
(x₁, y₁) = (-2, 2)
(x₂, y₂) = (-8, 6)
Midpoint = ((-2 + -8)/2, (2 + 6)/2)
= (-10/2, 8/2)
= (-5, 4)
Now we have the center coordinates: (-5, 4).
Next, let's calculate the distance from the center to one of the points, which will give us the radius:
Radius = √((x₂ - x₁)² + (y₂ - y₁)²)
Given points:
(x₁, y₁) = (-2, 2)
(x₂, y₂) = (-8, 6)
Radius = √((-8 - (-2))² + (6 - 2)²)
= √((-8 + 2)² + (6 - 2)²)
= √((-6)² + 4²)
= √(36 + 16)
= √52
= 2√13
So the radius of the circle is 2√13.
Now we have the center coordinates and the radius, we can write the equation of the circle:
Equation of a circle = (x - h)² + (y - k)² = r²
Where (h, k) is the center coordinates and r is the radius.
Plugging in our values, the equation of the circle is:
(x - (-5))² + (y - 4)² = (2√13)²
(x + 5)² + (y - 4)² = 4(13)
(x + 5)² + (y - 4)² = 52
Thus, the equation of the circle is (x + 5)² + (y - 4)² = 52.