Conic sections

A circle passes through points A(-5,2), B(-3,-4) and C(1,8). Find the coordinate of the point of intersection of the perpendicular bisectors of AB and BC. What is the equation of the circle?

Answer

To find the coordinates of the point of intersection of the perpendicular bisectors of AB and BC, we can follow these steps:

Step 1: Find the midpoint of AB and BC.
- The midpoint of a line segment is the average of the coordinates of its endpoints. We can use the midpoint formula for this:
Midpoint of AB = ((x1 + x2)/2, (y1 + y2)/2)

Midpoint of AB = ((-5 + (-3))/2, (2 + (-4))/2)
= (-8/2, -2/2)
= (-4, -1)

Midpoint of BC = ((x2 + x3)/2, (y2 + y3)/2)

Midpoint of BC = ((-3 + 1)/2, (-4 + 8)/2)
= (-2/2, 4/2)
= (-1, 2)

Step 2: Find the slopes of AB and BC.
- The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the slope formula:
Slope (m) = (y2 - y1)/(x2 - x1)

Slope of AB = (2 - (-4))/(-5 - (-3))
= (6)/(-2)
= -3

Slope of BC = (8 - (-4))/(1 - (-3))
= (12)/(4)
= 3

Step 3: Find the slopes of the perpendicular bisectors of AB and BC.
- The slopes of perpendicular lines are negative reciprocals of each other. To find the slopes of the perpendicular bisectors, we invert the signs of the slopes and flip the fraction:
Slope of perpendicular bisector of AB = -(-1/3)
= 1/3

Slope of perpendicular bisector of BC = -3
= -3

Step 4: Use the midpoint and slopes to find the equations of the perpendicular bisectors.
- We can use the point-slope form of a line equation to find the equations of the perpendicular bisectors. This form is given as:
y - y1 = m(x - x1)

For the perpendicular bisector of AB:
Equation of perpendicular bisector of AB: y - (-1) = (1/3)(x - (-4))
: y + 1 = (1/3)(x + 4)

For the perpendicular bisector of BC:
Equation of perpendicular bisector of BC: y - 2 = -3(x - (-1))
: y - 2 = -3(x + 1)

Step 5: Find the point of intersection of the perpendicular bisectors.
- To find the point of intersection between two lines, we can solve the system of equations formed by the equations of the perpendicular bisectors.

By setting the two equations equal to each other, we can find the x-coordinate:
(1/3)(x + 4) = -3(x + 1)

Solving this equation, we get:
x = -2

Substituting the value of x into either equation, we can find the y-coordinate:
y + 1 = (1/3)(-2 + 4)
y + 1 = (1/3)(2)
y + 1 = 2/3
y = 2/3 - 1
y = -1/3

Therefore, the point of intersection of the perpendicular bisectors of AB and BC is (-2, -1).

Step 6: Find the equation of the circle passing through points A, B, and C.
- Now that we know the center of the circle, which is the point of intersection of the perpendicular bisectors, we can find the radius by finding the distance between the center and any of the given points (A, B, or C).

Using the distance formula, the radius of the circle can be found:
Radius (r) = √((x - x1)^2 + (y - y1)^2)

Using point A(-5, 2):
r = √((-2 - (-5))^2 + (-1 - 2)^2)
r = √(3^2 + (-3)^2)
r = √(9 + 9)
r = √(18)

Therefore, the radius of the circle is √18.

The equation of a circle can be written as:
(x - h)^2 + (y - k)^2 = r^2, where (h, k) are the coordinates of the center.

Substituting the values into the equation, we get:
(x - (-2))^2 + (y - (-1))^2 = (√18)^2
(x + 2)^2 + (y + 1)^2 = 18

So, the equation of the circle passing through points A(-5,2), B(-3,-4), and C(1,8) is (x + 2)^2 + (y + 1)^2 = 18.

Hint: the point of intersection of the perpendicular bisectors is the center of the circle. Write the equations for the bisectors and solve for the point of intersection. Use the distance from any point to the center to get the radius and, from that, the circle's equation.

I gave you a method. Now you do the work.