Find the equation of the circle for which a diameter has endpoints P1(-1, 4) and P2(-5, 2).
centre would be the midpoint of the the two given points, which would be C(-3,3)
so the equation is
(x+3)^2 + (y-3)^2 = r^2
plug in one of the points given to find r^2
To find the equation of a circle when you have the endpoints of a diameter, you can use the distance formula. The distance formula is given by:
d = √((x2 - x1)² + (y2 - y1)²)
In this case, P1(-1, 4) and P2(-5, 2) are the endpoints of the diameter. So, we can use these points to find the center of the circle.
1. Find the midpoint of the diameter:
- To find the x-coordinate of the center, find the average of the x-coordinates of the endpoints: (x1 + x2)/2
Center_x = (-1 + -5)/2
= -6/2
= -3
- To find the y-coordinate of the center, find the average of the y-coordinates of the endpoints: (y1 + y2)/2
Center_y = (4 + 2)/2
= 6/2
= 3
2. Once you have the center of the circle, you need to find the radius. The radius is half the length of the diameter. You can use the distance formula again to find the length of the diameter.
d = √((x2 - x1)² + (y2 - y1)²)
Diameter = √((-5 - -1)² + (2 - 4)²)
= √((-5 + 1)² + (2 - 4)²)
= √((-4)² + (-2)²)
= √(16 + 4)
= √20
Radius = Diameter/2
= √20/2
= √5
3. Now that you have the center (-3, 3) and the radius √5, you can write the equation of the circle in standard form:
(x - Center_x)² + (y - Center_y)² = Radius²
(x + 3)² + (y - 3)² = (√5)²
(x + 3)² + (y - 3)² = 5
Therefore, the equation of the circle is (x + 3)² + (y - 3)² = 5.