Circles Flashcards

The graph of x² + (y − k)² = 25 is tangent to the x-axis. What are the possible values of k?

Answer: k = ±5

The circle is tangent to the x-axis when the distance from center (0, k) to x-axis equals the radius: |k| = 5, so k = ±5.

A circle has circumference 16π. What is the area of a sector with arc length 4π?

Answer: 16π

Radius = 8. Fraction of circle = 4π/16π = 1/4. Sector area = (1/4)·π(8²) = (1/4)(64π) = 16π.

The center of a circle is at (3, 4) and the circle is tangent to the line x = −1. What is the radius?

Answer: 4

Distance from center (3, 4) to the line x = −1 is |3 − (−1)| = 4, which is the radius.

In a circle with center O, chord AB is perpendicular to radius OC at point D, where D is the midpoint of AB. If OC = 10 and OD = 6, what is AB?

Answer: Both B and C

AD = √(OA² − OD²) = √(100−36) = 8. AB = 16. Both B and C are equivalent (√64 × 2 = 16).

Which of the following points is inside the circle (x − 1)² + (y + 2)² = 20?

Answer: (4, 1)

For (4,1): (4−1)²+(1+2)² = 9+9 = 18 20. For (−2,2): 9+16=25>20. The first listed inside point is (4,1).

The arc of a circle has length 10π. The radius is 15. What is the central angle in degrees?

Answer: 120°

Central angle in radians = arc/r = 10π/15 = 2π/3. Converting: (2π/3)(180/π) = 120°.

A circle has equation (x − h)² + (y − k)² = r². If the circle passes through (0, 0), (6, 0), and (0, 8), what is r?

Answer: 5

The circumscribed circle of a right triangle with legs 6 and 8 (hypotenuse 10) has diameter equal to the hypotenuse. So r = 10/2 = 5.