Circle Equations and Properties

The video tutorial covers the topic of circles in geometry, focusing on the standard form of a circle's equation. It explains the components of the equation, such as the center and radius, and provides several examples to illustrate how to write and solve circle equations. The tutorial also includes graphing exercises and real-world applications, such as using circle equations to determine distances in seismograph readings.

What is the standard form of the equation of a circle with center (h, k) and radius r?

• x^2 + y^2 = r^2

• x - h + y - k = r

• x - h)^2 + (y - k)^2 = r^2

• x^2 + y^2 = r

If the center of a circle is at the origin (0, 0), what does the equation simplify to?

• x - h)^2 + (y - k)^2 = r^2

• x^2 + y^2 = r^2

• x^2 + y^2 = r

• x + y = r

Given a circle with center (3, 5) and radius 6, what is the standard form of its equation?

• (x - 3)^2 + (y - 5)^2 = 6

• (x + 3)^2 + (y + 5)^2 = 36

• (x - 3)^2 + (y - 5)^2 = 36

• (x + 3)^2 + (y + 5)^2 = 6

What happens to the equation when you have x - (-8)?

• It becomes x + 8

• It becomes x + (-8)

• It becomes x - 8

• It remains x - (-8)

How do you simplify y - 0 in the equation of a circle?

• It becomes 0

• It remains y - 0

• It becomes y^2

• It becomes y

What is the center of the circle given by the equation (x + 4)^2 + (y - 1)^2 = 25?

• (4, -1)

• (-4, -1)

• (-4, 1)

• (4, 1)

What is the radius of the circle given by the equation (x + 4)^2 + (y - 1)^2 = 25?

• 5

• 25

• 4

• 1

Given the center (-3, -1) and a point (0, 0) on the circle, what is the radius?

• 10

• sqrt(10)

• 1

• 3

What is the equation of a circle with center (5, 8) passing through (-15, -13)?

• (x - 5)^2 + (y - 8)^2 = 841

• (x + 5)^2 + (y + 8)^2 = 841

• (x - 5)^2 + (y - 8)^2 = 400

• (x + 5)^2 + (y + 8)^2 = 400

If the epicenter of an earthquake is 5 miles away from point (2, 1), what is the radius of the circle?

• 4

• 2

• 1

• 5