Circle Equations and Properties

This video tutorial covers the topic of circle equations, focusing on how to determine the equation of a circle given its center and radius, as well as how to find the center and radius from a given equation. It also explores how to determine circle equations that touch specific lines and pass through given points. The tutorial provides step-by-step examples and encourages viewers to engage with the material through practice problems.

What is the general form of a circle equation centered at (P, A, B) with radius R?

• X^2 + Y^2 = R^2

• (X - A)^2 + (Y - B)^2 = R^2

• X^2 + Y^2 = 2R

• (X + A)^2 + (Y + B)^2 = R^2

What is the significance of the term (X - A)^2 in the circle equation?

• It represents the y-coordinate of the center

• It represents the radius

• It represents the x-coordinate of the center

• It represents the diameter

If a circle is centered at (3, 4) with a radius of 5, what is its equation?

• (X + 3)^2 + (Y + 4)^2 = 5

• (X - 3)^2 + (Y - 4)^2 = 5

• (X + 3)^2 + (Y + 4)^2 = 25

• (X - 3)^2 + (Y - 4)^2 = 25

Which of the following is a correct method to simplify a circle equation?

• Divide all terms by the same number

• Multiply all terms by the same number

• Subtract the same number from all terms

• Add the same number to all terms

How do you find the center of a circle from the equation (X - 1)^2 + (Y - 2)^2 = 25?

• Center is (2, 1)

• Center is (1, 2)

• Center is (5, 5)

• Center is (0, 0)

What is the radius of a circle with the equation (X - 3)^2 + (Y + 1)^2 = 24?

• 6

• 2√6

• 4

• 5

Which formula is used to find a circle equation that touches a line?

• (X - A)^2 + (Y - B)^2 = (AX + BY + C)^2 / (A^2 + B^2)

• (X - A)^2 + (Y - B)^2 = R^2

• (X - A)^2 + (Y - B)^2 = (C + D)^2

• (X + A)^2 + (Y + B)^2 = R^2

What is the equation of a circle centered at (2, 3) that touches the x-axis?

• (X - 2)^2 + (Y - 3)^2 = 9

• (X - 2)^2 + (Y - 3)^2 = 4

• (X - 2)^2 + (Y - 3)^2 = 3

• (X - 2)^2 + (Y - 3)^2 = 1

How do you determine the radius of a circle that passes through the origin and is centered at (2, 3)?

• Calculate the distance from the center to the origin

• Use the midpoint formula

• Use the formula R = √(A^2 + B^2)

• Set R = A + B

What is the equation of a circle centered at (5, -1) passing through (-1, 7)?

• (X + 5)^2 + (Y - 1)^2 = 100

• (X - 5)^2 + (Y + 1)^2 = 100

• (X + 5)^2 + (Y - 1)^2 = 64

• (X - 5)^2 + (Y + 1)^2 = 64