Circle Equations and Properties

This lesson covers the equations of circles, both centered and not centered at the origin. It explains the relationship between any point on the circle and its radius, using the Pythagorean theorem. The lesson includes practice problems for writing circle equations and determining if a point is inside, outside, or on a circle. It also covers finding the equation of a circle not centered at the origin and calculating the shortest distance from a point to a circle.

What is the definition of a circle?

• A shape with four equal sides

• A line segment with two endpoints

• A polygon with three sides

• A set of points equidistant from a center point

What is the equation of a circle centered at the origin?

• x^2 + y^2 = r^2

• x^2 + y^2 = 0

• x^2 + y^2 = 1

• x^2 + y^2 = x + y

Why does the equation x^2 + y^2 = r^2 describe a circle?

• It is a random equation

• It represents a square

• It shows the sum of distances

• It relates the radius to any point on the circle

How do you write the equation of a circle with a radius of 3 centered at the origin?

• x^2 + y^2 = 3

• x^2 + y^2 = 9

• x^2 + y^2 = 1

• x^2 + y^2 = 6

If the equation of a circle is x^2 + y^2 = 36, what is the radius?

• 12

• 18

• 36

• 6

A circle passes through the point (5, 3) and is centered at the origin. What is the equation of the circle?

• x^2 + y^2 = 16

• x^2 + y^2 = 34

• x^2 + y^2 = 25

• x^2 + y^2 = 9

How can you determine if a point is inside a circle?

• Check if x^2 + y^2 is not equal to the radius squared

• Check if x^2 + y^2 is greater than the radius squared

• Check if x^2 + y^2 is equal to the radius squared

• Check if x^2 + y^2 is less than the radius squared

What is the equation of a circle with center (3, 4) and radius 8?

• (x - 3)^2 + (y - 4)^2 = 64

• (x + 3)^2 + (y + 4)^2 = 64

• (x + 3)^2 + (y + 4)^2 = 8

• (x - 3)^2 + (y - 4)^2 = 8

What is the shortest distance from the point (10, 7) to the circle x^2 + y^2 = 49?

• 12 units

• 10 units

• 5.21 units

• 7 units