Understanding the Standard Equation of a Circle

This video tutorial explains how to find the standard equation of a circle given the endpoints of its diameter. It covers determining the center of the circle using the midpoint formula and calculating the radius by finding the distance between the endpoints. The tutorial then demonstrates how to derive the standard equation of the circle using these values.

What is the standard equation of a circle?

• x^2 + y^2 = 2r

• (x - h)^2 + (y - k)^2 = r^2

• (x + h)^2 + (y + k)^2 = r^2

• x^2 + y^2 = r^2

How do you find the center of a circle given the endpoints of its diameter?

• By adding the x and y coordinates

• By subtracting the y-coordinates

• By finding the midpoint of the diameter

• By finding the average of the x-coordinates

What is the x-coordinate of the center if the endpoints of the diameter are (-2, -3) and (-1, 5)?

• -3

• -0.5

• -1.5

• -2

What is the significance of the midpoint in the context of a circle's diameter?

• It is the radius of the circle

• It is the center of the circle

• It is the endpoint of the circle

• It is the circumference of the circle

Why is it easier to find the radius by calculating the diameter first?

• Because the center's x-coordinate is a fraction

• Because the radius is twice the diameter

• Because the diameter is shorter

• Because the diameter is always a whole number

What is the formula used to calculate the distance between two points?

• d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

• d = (x2 - x1)^2 - (y2 - y1)^2

• d = (x2 + x1)^2 + (y2 + y1)^2

• d = (x2 - x1) + (y2 - y1)

What is the value of r squared if the radius is sqrt(65)/2?

• 65/4

• 65/2

• 32.5

• 65

How is the radius related to the diameter?

• The radius is the square of the diameter

• The radius is equal to the diameter

• The radius is half the diameter

• The radius is twice the diameter

What is the final form of the circle's equation if h = -3/2, k = 1, and r^2 = 65/4?

• (x + 3/2)^2 + (y - 1)^2 = 65/4

• (x - 3/2)^2 + (y + 1)^2 = 65/4

• (x - 3/2)^2 + (y - 1)^2 = 65/4

• (x + 3/2)^2 + (y + 1)^2 = 65/4

What operation is performed to find r squared from the radius?

• Squaring

• Subtraction

• Addition

• Multiplication