equation of circle passing through P(10,3), Q(11,8), and R(7,14).
To find the equation of a circle passing through three given points, we can use the fact that the center of the circle lies at the intersection of the perpendicular bisectors of any two chords.
Let's find the equations of the perpendicular bisectors:
1. Find the midpoints of the chords formed by the given points P, Q, and R.
Midpoint of PQ:
x-coordinate: (10 + 11) / 2 = 21 / 2 = 10.5
y-coordinate: (3 + 8) / 2 = 11 / 2 = 5.5
Midpoint of QR:
x-coordinate: (11 + 7) / 2 = 18 / 2 = 9
y-coordinate: (8 + 14) / 2 = 22 / 2 = 11
2. Find the slopes of the lines passing through the given points P, Q, and R.
Slope of PQ: (8 - 3) / (11 - 10) = 5 / 1 = 5
Slope of QR: (14 - 8) / (7 - 11) = 6 / -4 = -3/2
3. Find the negative reciprocal of the slopes obtained above to get the slopes of the perpendicular bisectors.
Slope of perpendicular bisector of PQ: -1 / 5 = -1/5
Slope of perpendicular bisector of QR: -1 / (-3/2) = 2/3
4. Use the slope-intercept form of a line (y = mx + b) to find the equations of the perpendicular bisectors, substituting the midpoint coordinates from step 1:
Equation of perpendicular bisector of PQ:
y - 5.5 = (-1/5)(x - 10.5)
Equation of perpendicular bisector of QR:
y - 11 = (2/3)(x - 9)
5. Find the point of intersection of the perpendicular bisectors. To do this, solve the simultaneous equations of the two perpendicular bisector lines.
Equation 1: y - 5.5 = (-1/5)(x - 10.5)
Simplifying, we get: 5x + y = 59.5 (Equation A)
Equation 2: y - 11 = (2/3)(x - 9)
Simplifying, we get: 2x - 3y = -13 (Equation B)
Solving equations A and B simultaneously, we find the center of the circle:
Multiply equation A by 3 and equation B by 5 to eliminate the y-terms:
15x + 3y = 178.5 (Equation C)
10x - 15y = -65 (Equation D)
Adding equations C and D, we get:
25x = 113.5
x = 113.5 / 25 = 4.54
Substituting x = 4.54 into equation A, we can find y:
5(4.54) + y = 59.5
y = 59.5 - 22.7
y = 36.8
Therefore, the center of the circle is approximately (4.54, 36.8).
6. Finally, we can find the radius of the circle by calculating the distance between the center point and any of the three given points (P, Q, or R). Let's use the distance formula with point P(10, 3):
Radius = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Radius = sqrt((10 - 4.54)^2 + (3 - 36.8)^2)
Radius = sqrt(30.8496 + 1052.64)
Radius ≈ sqrt(1083.4896)
Radius ≈ 32.92
Now that we have the center and radius of the circle, we can write the equation in the standard form:
(x - h)^2 + (y - k)^2 = r^2
Substituting the values we obtained:
(x - 4.54)^2 + (y - 36.8)^2 = (32.92)^2
Therefore, the equation of the circle passing through P(10,3), Q(11,8), and R(7,14) is approximately (x - 4.54)^2 + (y - 36.8)^2 = 1083.49.