Understanding the Equation of a Circle

The video tutorial explains how to derive the general equation of a circle using the Pythagorean Theorem. It starts by defining a circle with a center at (h, k) and radius r, and then uses the theorem to establish the relationship between the circle's points and its center. The tutorial provides a step-by-step derivation of the equation (x-h)^2 + (y-k)^2 = r^2 and includes an example to illustrate how to apply this formula when given specific values for the center and radius.

What is the center of a circle with the equation (x - h)^2 + (y - k)^2 = r^2?

• (h, k)

• (r, r)

• (0, 0)

• (x, y)

Which theorem is used to relate the distance between a point on the circle and its center?

• Pythagorean Theorem

• Binomial Theorem

• Fundamental Theorem of Calculus

• Fermat's Last Theorem

In the context of a circle, what does the hypotenuse of the right triangle represent?

• The area of the circle

• The circumference of the circle

• The diameter of the circle

• The radius of the circle

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

• x^2 + y^2 = 2r

• (x + h)^2 + (y + k)^2 = r^2

• x^2 + y^2 = r^2

• (x - h)^2 + (y - k)^2 = r^2

If a circle has a center at (5, -5) and a radius of 4, what is its equation?

• (x - 5)^2 + (y - 5)^2 = 16

• (x + 5)^2 + (y - 5)^2 = 16

• (x + 5)^2 + (y + 5)^2 = 16

• (x - 5)^2 + (y + 5)^2 = 16

How do you determine the center of a circle from its equation (x - h)^2 + (y - k)^2 = r^2?

• The center is (h, k)

• The center is (x, y)

• The center is (r, r)

• The center is (0, 0)

What happens to the equation of a circle if it is centered at the origin?

• It becomes (x + h)^2 + (y + k)^2 = r^2

• It becomes x^2 + y^2 = 2r

• It becomes (x - h)^2 + (y - k)^2 = r^2

• It becomes x^2 + y^2 = r^2

If the equation of a circle is given as (x - 3)^2 + (y + 2)^2 = 25, what is the radius?

• 2

• 3

• 5

• 25

In the equation (x - h)^2 + (y - k)^2 = r^2, what does r represent?

• The circumference of the circle

• The diameter of the circle

• The center of the circle

• The radius of the circle

What is the equation of a circle with a center at (0, 0) and radius 7?

• x^2 + y^2 = 7

• x^2 + y^2 = 49

• (x - 7)^2 + (y - 7)^2 = 49

• (x + 7)^2 + (y + 7)^2 = 49