Circles Flashcards

A circle has center (a, b) and is tangent to the x-axis. Which of the following must be true?

Answer: |b| = radius

A circle tangent to the x-axis has its distance to x-axis equal to the radius: |b| = radius, regardless of the sign of b.

In a circle, the ratio of arc length to radius is the central angle in what units?

Answer: Radians

By definition, arc length = rθ where θ is in radians, so θ = arc length / radius is measured in radians.

Circle P has equation x² + y² − 8x + 6y = 0. What is the sum of the coordinates of the center?

Answer: 1

Completing the square: (x−4)² − 16 + (y+3)² − 9 = 0 → (x−4)² + (y+3)² = 25. Center = (4, −3). Sum = 4 + (−3) = 1.

An arc in a circle with radius 8 has length 6π. What is the area of the sector formed by this arc?

Answer: 24π

Central angle θ = arc/r = 6π/8 = 3π/4 radians. Sector area = (1/2)r²θ = (1/2)(64)(3π/4) = 24π.

A chord of a circle subtends a central angle of 60° and has length 10. What is the radius of the circle?

Answer: 10

For a chord of length c and central angle θ: c = 2r·sin(θ/2). So 10 = 2r·sin(30°) = 2r·(0.5) = r, giving r = 10.

A circular track has a diameter of 400 meters. An athlete runs 3/4 of the way around the track. How far does the athlete run?

Answer: 300π meters

Circumference = π(400) = 400π. Distance = (3/4)(400π) = 300π meters.

Two circles have the same center. The inner circle has radius 4 and the outer circle has radius 7. A point is chosen at random inside the outer circle. What is the probability that it lies in the annular region between the two circles?

Answer: 33/49

Area of outer circle = 49π; area of inner circle = 16π; annulus area = 33π. P = 33π/49π = 33/49.