Exploring the Equation of a Circle

The video tutorial explains how to find and write the equation of a circle. It begins with a circle centered at the origin and uses the Pythagorean theorem to determine the radius. The tutorial then generalizes the equation for circles with centers not at the origin, demonstrating how to adjust the formula accordingly. The key formula for any circle is presented as (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.

What is the radius of a circle centered at (0,0) with points extending 5 units from the center?

• 3 units

• 4 units

• 5 units

• 6 units

How do you calculate the distance of a point from the center of a circle using coordinates?

• Subtracting y-coordinates

• Subtracting x-coordinates

• Using the Pythagorean theorem

• Multiplying the coordinates

What is the equation of a circle with a radius of 5 centered at (0,0)?

• x^2 + y^2 = 25

• x^2 + y^2 = 5

• x^2 - y^2 = 5

• x^2 - y^2 = 25

What does the 'c' represent in the circle equation x^2 + y^2 = c^2?

• Circumference of the circle

• Center of the circle

• Diameter of the circle

• Radius of the circle

How is the distance from the center to a point on the circle represented in the circle's equation?

• As the variable 'y'

• As the variable 'c'

• As the variable 'x'

• As the variable 'r'

For a circle with center at (2, -6), what is the correct way to calculate the x distance from a point?

• Add 2 to the point's x-coordinate

• Multiply the point's x-coordinate by 2

• Divide the point's x-coordinate by 2

• Subtract 2 from the point's x-coordinate

If a circle's center is moved from (0,0) to (2, -6), how does the equation change?

• The radius changes

• The sign of the center coordinates in the equation is inverted

• The equation remains the same

• The equation becomes linear

What is the effect of changing the center of a circle from (0,0) to any other point on its equation?

• The equation becomes a parabola

• The radius in the equation changes

• Subtraction terms are added to x and y in the equation

• The equation remains unchanged

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

• (x - h)^2 + (y - k)^2 = r

• (x + h)^2 + (y + k)^2 = r^2

• (x - h)^2 + (y - k)^2 = r^2

• (x + h)^2 - (y + k)^2 = r^2

In the general equation of a circle, what do 'h' and 'k' represent?

• The x and y intercepts

• The radius and diameter

• The slope and intercept

• The coordinates of the center