Understanding the Equation of a Circle

This video introduces the equation of a circle, explaining how to derive it using the center and radius. It provides examples with circles centered at (0,0), (5,2), and (-2,3), demonstrating how to write their equations. The video also shows how to verify if a point lies on a circle by substituting the point's coordinates into the circle's equation.

What is the general form of the equation of a circle?

• (x + a)^2 + (y + b)^2 = r^2

• x^2 + y^2 = 2r

• (x - a)^2 + (y - b)^2 = r^2

• x^2 + y^2 = r^2

In the equation (x - a)^2 + (y - b)^2 = r^2, what do 'a' and 'b' represent?

• The coordinates of a point on the circle

• The coordinates of the center of the circle

• The diameter of the circle

• The radius of the circle

What does the term (x - a)^2 + (y - b)^2 = r^2 represent in geometry?

• A line

• A parabola

• A circle

• An ellipse

If a circle is centered at (0, 0) with a radius of 5, what is its equation?

• (x + 5)^2 + (y + 5)^2 = 25

• x^2 + y^2 = 5

• (x - 5)^2 + (y - 5)^2 = 25

• x^2 + y^2 = 25

How do you modify the circle equation for a center at (5, 2)?

• (x + 5)^2 + (y + 2)^2 = r^2

• (x - 5)^2 + (y - 2)^2 = r^2

• (x + 2)^2 + (y + 5)^2 = r^2

• (x - 2)^2 + (y - 5)^2 = r^2

What is the radius of a circle if its equation is (x - 5)^2 + (y - 2)^2 = 4?

• 16

• 8

• 4

• 2

What is the equation of a circle with center (-2, 3) and radius 1?

• (x - 2)^2 + (y + 3)^2 = 1

• (x - 2)^2 + (y - 3)^2 = 1

• (x + 2)^2 + (y + 3)^2 = 1

• (x + 2)^2 + (y - 3)^2 = 1

If a circle's equation is (x + 2)^2 + (y - 3)^2 = 1, what is the radius?

• 4

• 3

• 2

• 1

How do you verify if a point lies on a given circle?

• Calculate the distance from the point to the center and compare with the radius

• Check if the point is the center of the circle

• Draw the circle and see if the point is on the circumference

• Substitute the point into the circle's equation and check if the equation holds

What is the result when substituting the point (-2, 2) into the equation (x - 1)^2 + (y - 2)^2 = 3^2?

• The point does not satisfy the equation

• The point lies inside the circle

• The point lies on the circle

• The point lies outside the circle