1.How do I find the center of a circle using the points (2,8) (-4,6) (1,-1) ?
2. The Radius of the circle?
3. Equation of circle
THANKS!
1. To find the center of a circle using three points, you can use the method of finding the intersection of the perpendicular bisectors of any two line segments formed by the three given points. Here's how:
- Step 1: Take two of the given points, say (2,8) and (-4,6). To find the midpoint of the line segment connecting these two points, use the midpoint formula:
Midpoint X-coordinate = (X1 + X2)/2 = (2 + (-4))/2 = -1
Midpoint Y-coordinate = (Y1 + Y2)/2 = (8 + 6)/2 = 7
- Step 2: Determine the slope of the line segment connecting the two points using the slope formula:
Slope = (Y2 - Y1)/(X2 - X1) = (6 - 8)/(-4 - 2) = -2/-6 = 1/3
- Step 3: Find the negative reciprocal of the slope calculated in the previous step to obtain the slope of the perpendicular bisector:
Negative reciprocal = -1/(1/3) = -3
- Step 4: Using the midpoint and the slope of the perpendicular bisector, write the equation of the perpendicular bisector in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Substituting the values, we get:
7 = (-3)(-1) + b
7 = 3 + b
b = 7 - 3 = 4
- Step 5: The equation of the perpendicular bisector is y = -3x + 4. Repeat steps 1-4 with different combinations of the given points to find two more equations of perpendicular bisectors.
- Step 6: Solve the system of two equations, obtained in step 5, for the intersection point which is the center of the circle. You can do this by setting the equations equal to each other and solving for both x and y values.
2. To find the radius of the circle using the center and any of the given points, you can use the distance formula. Here's how:
- Select a point from the given points, let's say (2,8), and use the distance formula to find the distance between this point and the center of the circle:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((2 - center_x)^2 + (8 - center_y)^2)
3. To find the equation of the circle, you can use the formula:
(x - center_x)^2 + (y - center_y)^2 = radius^2
By plugging in the values of the center (obtained in step 6) and the radius (obtained in step 2) into this formula, you can find the equation of the circle.