Finding the Equation of a Circle from Three Points

This video tutorial explains how to find the equation of a circle passing through three points. It covers the necessary prerequisites, such as writing the equation of a circle, finding the distance between two points, and understanding perpendicular slopes. The tutorial demonstrates how to find the center of the circle using perpendicular bisectors of chords formed by the three points. An example problem is provided to illustrate the process, including calculating the center and radius of the circle and writing its equation.

What is the general equation of a circle with center (h, k) and radius R?

• (x - h)^2 + (y - k)^2 = R

• x^2 + y^2 = R^2

• x^2 + y^2 = R

• (x - h)^2 + (y - k)^2 = R^2

How do you find the distance between two points (x1, y1) and (x2, y2)?

• sqrt((x2 - x1)^2 + (y2 - y1)^2)

• sqrt((x2 + x1)^2 + (y2 + y1)^2)

• |x2 - x1| + |y2 - y1|

• (x2 - x1)^2 + (y2 - y1)^2

What is the slope of a line perpendicular to another line with slope m?

• 1/m

• -1/m

• -m

• -1/m^2

How do you find the midpoint between two points (x1, y1) and (x2, y2)?

• (x1 + x2, y1 + y2)

• (x1 * x2, y1 * y2)

• ((x1 + x2) / 2, (y1 + y2) / 2)

• ((x1 - x2) / 2, (y1 - y2) / 2)

What property do the perpendicular bisectors of chords in a circle have?

• They intersect at the circle's center

• They are parallel to the circle's radius

• They bisect the circle's area

• They form a right angle with the circle's diameter

How is the equation of the perpendicular bisector of a segment derived?

• Using the midpoint and the slope of the segment

• Using the endpoints of the segment only

• Using the slope of the segment and one endpoint

• Using the midpoint and the perpendicular slope

What method is used to solve the system of equations derived from the perpendicular bisectors?

• Graphical method

• Substitution

• Matrix method

• Addition or subtraction

How do you determine the radius of the circle?

• By measuring the circumference of the circle

• By doubling the distance between the center and one of the points

• By finding the distance from the center to any point on the circle

• By calculating the area of the circle

What is the center of the circle found in the example?

• (-8, -1)

• (8, 1)

• (1, 8)

• (8, -1)

What is the final form of the equation of a circle found in the example?

• (x - 8)^2 + (y - 1)^2 = sqrt(29)

• (x + 8)^2 + (y + 1)^2 = 29

• x^2 + y^2 = 29

• (x - 8)^2 + (y - 1)^2 = 29