Understanding Cartesian and Polar Equations of a Circle

The video tutorial explains how to derive both Cartesian and polar equations for a circle centered at the origin with a radius of 5 units. It begins by deriving the Cartesian equation using the standard formula and substituting the center and radius values. The tutorial then presents two methods for finding the polar equation: the first involves recognizing the relationship between Cartesian and polar coordinates, while the second uses trigonometric identities. Both methods result in the polar equation r = 5, with explanations of the steps involved.

What is the center of the circle described in the problem?

• (0, 0)

• (10, 10)

• (5, 5)

• (1, 1)

What is the radius of the circle in the given problem?

• 3 units

• 10 units

• 5 units

• 7 units

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

• (x - h)^2 + (y - k)^2 = r

• (x - h)^2 + (y - k)^2 = r^2

• x^2 + y^2 = r

• x^2 + y^2 = r^2

What is the simplified Cartesian equation for the circle centered at the origin with radius 5?

• x^2 + y^2 = 25

• x^2 + y^2 = 50

• x^2 + y^2 = 10

• x^2 + y^2 = 5

In the first method for finding the polar equation, what is substituted for x^2 + y^2?

• r

• r^2

• theta

• cos(theta)

What is the polar equation derived from the first method?

• r = 10

• r = 25

• r = ±5

• r = 5

In the second method, what is x replaced with?

• r sin(theta)

• r

• r cos(theta)

• r tan(theta)

What identity is used in the second method to simplify the equation?

• Pythagorean identity

• Trigonometric identity

• Logarithmic identity

• Exponential identity

What is the common form of the polar equation for the circle?

• r = 10

• r = 25

• r = 5

• r = ±5

Which of the following is true about the polar equations derived?

• Both r = 5 and r = -5 are correct

• Only r = -5 is correct

• Neither r = 5 nor r = -5 is correct

• Only r = 5 is correct