Solving Word Problems Involving Circles (3.4.4)

Question 1

Find all points where x = y (i.e., the point’s x-coordinate is
equal to its y-coordinate) that are also 4 units from (1, 3).
All points 4 units from (1, 3) are on the circle with center
(1, 3) and radius 4.
Use the coordinates of the center, (1, 3), and the radius, 4, to
write the equation of the circle.
(x – h)
2
+ (y – k)
2
= r
2
(x – 1)2
+ (y – 3)2
= 16
To find all points on (x – 1)2
+ (y – 3)2
= 16 where x = y,
substitute x into the equation of the circle for y.
(x – 1)2
+ (y – 3)2
= 16
(x – 1)2
+ (x – 3)2
= 16
Now simplify by expanding the powers of the binomials
(foiling) and combining the like terms. Then subtract 16 from
each side to get 0 on one side.
The resulting trinomial cannot be factored. So, use the
quadratic formula to solve for x.
Notice that the quadratic formula gave two values for x.
Therefore, there are two points on the circle where x = y.
Since the x-coordinate must be equal to the y-coordinate in
each point, no calculations are needed to find each point’s
y-coordinate. Just repeat the x-coordinate as the y-coordinate
in each point.
Notice that these points are the points of intersection between
the circle defined by (x – 1)2
+ (y – 3)2
= 16 and the line
defined by x = y

Answer 1

Find all points where x = y (i.e., the point’s x-coordinate is
equal to its y-coordinate) that are also 4 units from (1, 3).
All points 4 units from (1, 3) are on the circle with center
(1, 3) and radius 4.
Use the coordinates of the center, (1, 3), and the radius, 4, to
write the equation of the circle.
(x – h)
2
+ (y – k)
2
= r
2
(x – 1)2
+ (y – 3)2
= 16
To find all points on (x – 1)2
+ (y – 3)2
= 16 where x = y,
substitute x into the equation of the circle for y.
(x – 1)2
+ (y – 3)2
= 16
(x – 1)2
+ (x – 3)2
= 16
Now simplify by expanding the powers of the binomials
(foiling) and combining the like terms. Then subtract 16 from
each side to get 0 on one side.
The resulting trinomial cannot be factored. So, use the
quadratic formula to solve for x.
Notice that the quadratic formula gave two values for x.
Therefore, there are two points on the circle where x = y.
Since the x-coordinate must be equal to the y-coordinate in
each point, no calculations are needed to find each point’s
y-coordinate. Just repeat the x-coordinate as the y-coordinate
in each point.
Notice that these points are the points of intersection between
the circle defined by (x – 1)2
+ (y – 3)2
= 16 and the line
defined by x = y

Question 2

There are many circles that pass through the points (1, 4) and
(–3, 2). The circle with the smallest radiusthat passes through
these points must have a diameter with endpoints at (1, 4)
and (–3, 2).
If (1, 4) and (–3, 2) are the endpoints of a diameter, then the
circle’s center must be at the midpoint of (1, 4) and (–3, 2).
Use the midpoint formula to find the circle’s center.
The circle’s radius is the distance between its center and any
point on the circle. So, find the radius by using the distance
formula to find the distance between the center and either
(1, 4) or (–3, 2).
Once the center and radius are known, just substitute into the standard equation of a circle to write the circle’s equation.

Answer 2

There are many circles that pass through the points (1, 4) and
(–3, 2). The circle with the smallest radiusthat passes through
these points must have a diameter with endpoints at (1, 4)
and (–3, 2).
If (1, 4) and (–3, 2) are the endpoints of a diameter, then the
circle’s center must be at the midpoint of (1, 4) and (–3, 2).
Use the midpoint formula to find the circle’s center.
The circle’s radius is the distance between its center and any
point on the circle. So, find the radius by using the distance
formula to find the distance between the center and either
(1, 4) or (–3, 2).
Once the center and radius are known, just substitute into the standard equation of a circle to write the circle’s equation.