A. determine the equation of the circle passing through points A(3;2) B(8;12) and C(-6;-2)
B. a chord of the circle x^2+y^2=50 has a midpoint at (6;3). determine the equation of this chord.
A. To determine the equation of a circle passing through three points, we need to find the center and the radius of the circle.
1. Find the equations of the perpendicular bisectors of two of the three line segments formed by the given points. Let's take points A and B.
a. Find the midpoint of AB: M(ab) = ((x1 + x2)/2, (y1 + y2)/2)
M(ab) = ((3 + 8)/2, (2 + 12)/2) = (11/2, 7)
b. Find the slope of AB: m(ab) = (y2 - y1) / (x2 - x1)
m(ab) = (12 - 2) / (8 - 3) = 10/5 = 2
c. The slope of the perpendicular bisector is the negative reciprocal of m(ab). So, the slope of the perpendicular bisector will be -1/2.
d. Find the equation of the perpendicular bisector using the midpoint and slope:
y - y1 = m(perpendicular bisector) * (x - x1)
y - 7 = -1/2 * (x - 11/2)
2y - 14 = -x + 11
x + 2y = 25
2. Repeat the steps above for points A and C.
a. Find the midpoint of AC: M(ac) = ((x1 + x2)/2, (y1 + y2)/2)
M(ac) = ((3 - 6)/2, (2 - 2)/2) = (-3/2, 0)
b. Find the slope of AC: m(ac) = (y2 - y1) / (x2 - x1)
m(ac) = (0 - 2) / (-6 - 3) = -2/-9 = 2/9
c. The slope of the perpendicular bisector is the negative reciprocal of m(ac). So, the slope of the perpendicular bisector will be -9/2.
d. Find the equation of the perpendicular bisector using the midpoint and slope:
y - y1 = m(perpendicular bisector) * (x - x1)
y - 0 = -9/2 * (x + 3/2)
2y = -9x - 27/2
9x + 2y = -27
3. Solve the system of equations formed by the two perpendicular bisectors:
x + 2y = 25
9x + 2y = -27
Subtracting the first equation from the second equation, we get:
9x + 2y - (x + 2y) = -27 - 25
8x = -52
x = -52/8 = -13/2
Substituting the value of x into the first equation, we get:
(-13/2) + 2y = 25
2y = 25 + 13/2
2y = 50/2 + 13/2
2y = 63/2
y = 63/4 = 15.75
4. Now, we have the center of the circle, which is (-13/2, 15.75). To find the radius, we can use the distance formula between the center and one of the given points, such as A(3, 2):
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
r = sqrt((-13/2 - 3)^2 + (15.75 - 2)^2)
r = sqrt((-13/2 - 6)^2 + (15.75 - 2)^2)
r = sqrt((-32/2)^2 + (13.75)^2)
r = sqrt(256 + 189.0625)
r = sqrt(445.0625)
r ≈ 21.10
Therefore, the equation of the circle passing through points A(3,2), B(8,12), and C(-6,-2) is (x + 6.5)^2 + (y - 15.75)^2 ≈ 445.06.
B. To determine the equation of a chord, we need two points on the chord.
1. We know that the midpoint of the chord is (6, 3). Let's call the other point on the chord (x₁, y₁).
2. The equation of the circle is given as x^2 + y^2 = 50.
3. Use the midpoint formula to find the coordinates of the other point on the chord:
x₁ = 2 * x(midpoint) - x₂
x₁ = 2 * 6 - x₂
x₁ = 12 - x₂
y₁ = 2 * y(midpoint) - y₂
y₁ = 2 * 3 - y₂
y₁ = 6 - y₂
4. Substitute these values into the equation of the circle:
(12 - x₂)^2 + (6 - y₂)^2 = 50
Simplify the equation:
144 - 24x₂ + x₂^2 + 36 - 12y₂ + y₂^2 = 50
Combine like terms:
x₂^2 + y₂^2 - 24x₂ - 12y₂ + 130 = 0
Therefore, the equation of the chord of the circle x² + y² = 50 with a midpoint at (6, 3) is x² + y² - 24x - 12y + 130 = 0.