Mastering Circle Equations

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

• (x + h)² + (y + k)² = r²

• x² + y² = r²

• (x - h)² + (y - k)² = r²

• (x - h)(y - k) = r²

Given the center (3, -2) and radius 5, write the equation of the circle.

• (x + 3)² + (y - 2)² = 20

• (x - 3)² + (y + 2)² = 30

• (x + 3)² + (y + 2)² = 25

• (x - 3)² + (y + 2)² = 25

Convert the equation x^2 + y^2 - 6x + 4y - 12 = 0 to standard form.

• (x + 3)^2 + (y - 2)^2 = 16

• (x + 3)^2 + (y + 2)^2 = 30

• (x - 3)^2 + (y - 2)^2 = 20

• (x - 3)^2 + (y + 2)^2 = 25

Find the center and radius of the circle given by (x - 1)^2 + (y + 3)^2 = 16.

• Center: (1, -2), Radius: 6

• Center: (1, -3), Radius: 4

• Center: (2, -3), Radius: 3

• Center: (0, -4), Radius: 5

Determine if the point (2, 1) lies on the circle x^2 + y^2 = 10.

• It lies on the circle

• The point is outside the circle

• No

• Yes

Write the equation of a circle with center at (0, 0) and radius 7.

• x² + y² = 49

• x² + y² = 14

• (x - 7)² + (y - 7)² = 0

• x² + y² = 56

If the equation of a circle is (x + 4)^2 + (y - 1)^2 = 25, what is the center?

• (-4, 1)

• (-4, -1)

• (4, -1)

• (0, 1)

Find the radius of the circle defined by the equation (x - 2)^2 + (y + 5)^2 = 36.

• 4

• 8

• 6

• 10

What is the distance between the centers of the circles (x - 1)^2 + (y - 2)^2 = 9 and (x + 3)^2 + (y + 4)^2 = 16?

• 3sqrt(5)

• 10

• 5

• 2sqrt(13)

How do you determine if two circles intersect based on their equations?

• The circles intersect if |r1 - r2| > d.

• The circles intersect if |r1 - r2| < d < r1 + r2.

• The circles intersect if d > r1 + r2.

• The circles intersect if d = r1 + r2.