Question

point P=(8,17) and Q=(-2,17) are the endpoints of the diameter of a circle
a) what is the equation of the circle
b) what is the length of the arc of the circle between Point P and Point R=(0,13)
c) what is the angle formed by a sector whose area is 25/12pi

Answer 1

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Answer:

  a) (x -3)^2 +(y -17)^2 = 25

  b) ≈ 11.07

  c) π/6 radians

Step-by-step explanation:

a) The center of the circle is the midpoint of the given segment, so is ...

  A = (P +Q)/2 = ((8, 17) +(-2, 17))/2 = (6, 34)/2 = (3, 17) . . . center

The radius is the difference between center and end coordinates: 8 -3 = 5.

The equation of a circle with center (h, k) and radius r is ...

  (x -h)^2 +(y -k)^2 = r^2

This circle's equation is ...

  (x -3)^2 +(y -17)^2 = 25

__

b) The length of the arc is given by ...

  s = rθ

where r is the circle radius and θ is the central angle in radians.

We can find the central angle using a little trigonometry. Point R differs from point A by ...

  R -A = (0, 13) -(3, 17) = (-3, -4)

so is 4 units below and 3 units left of the center of the circle. The sine of this angle QAR is 4/5, so its value is ...

  angle QAR = arcsin(4/5) ≈ 53.13°

Then angle PAR is the supplement of this, so about 126.87°, or 2.2143 radians. Then the arc length is ...

  s = (5)(2.2143 radians)

  s ≈ 11.07

__

c) The area of a sector is given by ...

  A = (1/2)r^2·θ

Then the angle is ...

  θ = (2A)/r^2

For the given values and r=5, we have ...

  θ = 2(25/12π)/25

  θ = π/6

The sector with that area will have a central angle of π/6 radians, or 30°.

Answer 2

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a) (x-3)2 +(y-17) 2 = 25

b) 11.07

c) T7/6 radians

Step-by-step explanation:

a) The center of the circle is the midpoint

of the given segment, so is ...

A (P +Q)/2 = (8, 17) +(-2, 17)/2 = (6, 34)/

2= (3, 17)... center

center and end coordinates: 8-3 = 5.

The radius is the difference between

The equation of a circle with center (h, k)

and radius r is ...

x-h 2 + -k^2 = r^2

(x-3)2 +(y -17) 2 = 25

This circle's equation is

________

b) The length of the arc is given by ..

S = re

where r is the circle radius and e is the

We can find the central angle using a little

trigonometry. Point R differs from point A

by

R-A (0, 13)-(3, 17) = (-3,-4)

center of the circle. The sine of this angle

QAR is 4/5, so its value is..

so is 4 units below and 3 units left of the

angle QAR = arcsin(4/5) = 53.13

Then angle PAR is the supplement of this,

so about 126.87 , or 2.2143 radians. Then

the arc length is..

S = (5)(2.2143 radians)

s 11.07

________

c) The area of a sector is given by ..

A = (1/2)r2.0

Then the angle is ..

= (2A)/2

For the given values and r=5, we have..

2(25/12m)/25

T/6

central angle of m/6 radians, or 30°.

The sector with that area will have a