Page 1
Question ID: c8345903
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
The circle above has center O, the length of arc is , and . What is the length of arc ?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the
arcs. It’s given that arc is subtended by a central angle with measure 100°. Since the sum of the measures of the angles about a point is
360°, it follows that arc is subtended by a central angle with measure . If s is the length of arc , then s
must satisfy the ratio . Reducing the fraction to its simplest form gives . Therefore, . Multiplying both
sides of by yields .
Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc is greater than half of the circle. Choice C
is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc , not its full length.
Page 2
Question ID: 35d37640
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
Point lies on a unit circle in the xy-plane and has coordinates . Point is the center of the circle and has coordinates . Point
also lies on the circle and has coordinates , where is a constant. Which of the following could be the positive measure of angle ,
in radians?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. It's given that the circle is a unit circle, which means the circle has a radius of . It's also given that point is the center of
the circle and has coordinates
and that point lies on the circle and has coordinates . Since the radius of the circle is , the value
of must be , as all other points with an x-coordinate of are a distance greater than away from point . Since and are points on the
unit circle centered at , let side be the starting side of the angle and side be the terminal side of the angle. Since side is on the
positive x-axis and side is on the negative x-axis, side is half of a rotation around the unit circle, or radians, away from side .
Therefore, the positive measure of angle , in radians, is equal to plus an integer multiple of . In other words, the positive measure of
angle , in radians, is an odd integer multiple of . Of the given choices, only is an odd integer multiple of .
Choice A is incorrect. This could be the positive measure of an angle where the starting side is and the terminal side contains the point
, not .
Choice B is incorrect. This could be the positive measure of an angle where the starting side is and the terminal side contains the point
, not .
Choice C is incorrect. This could be the positive measure of an angle where the starting side is and the terminal side contains the point
, not .
Page 3
Question ID: 2266984b
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. The standard equation of a circle in the xy-plane is of the form , where
are the coordinates
of the center of the circle and r is the radius. The given equation can be rewritten in standard form by completing the squares. So the sum of the
first two terms, , needs a 100 to complete the square, and the sum of the second two terms, , needs a 64 to complete the
square. Adding 100 and 64 to both sides of the given equation yields , which is
equivalent to . Therefore, the coordinates of the center of the circle are .
Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete the squares or when identifying
the coordinates of the center.
Page 4
Question ID: 69b0d79d
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
Point O is the center of the circle above, and the measure of is . If the length of is 18, what is the length of arc ?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. Because segments OA and OB are radii of the circle centered at point O, these segments have equal lengths. Therefore,
triangle AOB is an isosceles triangle, where angles OAB and OBA are congruent base angles of the triangle. It’s given that angle OAB measures
. Therefore, angle OBA also measures . Let represent the measure of angle AOB. Since the sum of the measures of the three
angles of any triangle is , it follows that , or . Subtracting from both sides of this
equation yields , or radians. Therefore, the measure of angle AOB, and thus the measure of arc , is radians. Since
is a radius of the given circle and its length is 18, the length of the radius of the circle is 18. Therefore, the length of arc can be
calculated as , or .
Choices A, C, and D are incorrect and may result from conceptual or computational errors.
Page 5
Question ID: b8a225ff
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
Circle A in the xy-plane has the equation . Circle B has the same center as circle A. The radius of circle B is two times
the radius of circle A. The equation defining circle B in the xy-plane is , where is a constant. What is the value of ?
Correct Answer: 16
Rationale
The correct answer is . An equation of a circle in the xy-plane can be written as , where the center of the circle is
, the radius of the circle is , and where , , and are constants. It’s given that the equation of circle A is , which
is equivalent to . Therefore, the center of circle A is
and the radius of circle A is . It’s given that circle B has
the same center as circle A and that the radius of circle B is two times the radius of circle A. Therefore, the center of circle B is
and the
radius of circle B is
, or . Substituting for , for , and for into the equation
yields
, which is equivalent to . It follows that the equation of circle B in the xy-plane is
. Therefore, the value of is .
Page 6
Question ID: ab176ad6
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
The equation defines a circle in the xy‑plane. What is the radius of the circle?
Rationale
The correct answer is 11. A circle with equation , where a, b, and r are constants, has center and radius r.
Therefore, the radius of the given circle is , or 11.
Page 7
Question ID: 3e577e4a
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
A circle in the xy-plane has its center at . Line is tangent to this circle at the point . What is the slope of line ?
Answer
A.
B.
C.
D.
Correct Answer: A
Rationale
Choice A is correct. A line that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It's given that the circle has
its center at and line is tangent to the circle at the point . The slope of a radius defined by the points and can
be calculated as . The points and define the radius of the circle at the point of tangency. Therefore, the slope of this
radius can be calculated as
, or . If a line and a radius are perpendicular, the slope of the line must be the negative reciprocal of the
slope of the radius. The negative reciprocal of
is . Thus, the slope of line is.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the slope of the radius of the circle at the point of tangency, not the slope of line .
Choice D is incorrect and may result from conceptual or calculation errors.
Page 8
Question ID: fa2771d5
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
Circle A has equation . In the -plane, circle B is obtained by translating circle A to the right units. Which equation
represents circle B?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. The equation of a circle in the xy-plane can be written as
, where the center of the circle is
and the radius of the circle is units. It’s given that circle A has the equation , which can be written as
. It follows that , , and . Therefore, the center of circle A is and its radius is unit. If circle A is
translated
units to the right, the x-coordinate of the center will increase by , while the y-coordinate and the radius of the circle will remain
unchanged. Translating the center of circle A to the right units yields , or . Therefore, the center of circle B is .
Substituting
for , for , and for into the equation
yields , or
. Therefore, the equation
represents circle B.
Choice A is incorrect. This equation represents a circle obtained by shifting circle A down, rather than right, units.
Choice B is incorrect. This equation represents a circle obtained by shifting circle A left, rather than right, units.
Choice D is incorrect. This equation represents a circle obtained by shifting circle A up, rather than right, units.
Page 9
Question ID: 8e7689e0
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
The number of radians in a 720-degree angle can be written as , where a is a constant. What is the value of a ?
Rationale
The correct answer is 4. There are radians in a angle. An angle measure of is 4 times greater than an angle measure of .
Therefore, the number of radians in a angle is .
Page 10
Question ID: f2495de4
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Easy
Question
What is the value of
?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. The cosine of an angle is equal to the cosine of radians more than the angle, where is an integer constant. Since
is equivalent to , can be rewritten as , which is equal to . Therefore, the value of
is equal to the value of , which is .
Alternate approach: A trigonometric ratio can be found using the unit circle, that is, a circle with radius unit. The cosine of a number is the x-
coordinate of the point resulting from traveling a distance of counterclockwise from the point around a unit circle centered at the origin
in the xy-plane. A unit circle has a circumference of . It follows that since is equal to , traveling a distance of
counterclockwise around a unit circle means traveling around the circle completely times and then another beyond that. That is, traveling
results in the same point as traveling . Traveling counterclockwise from the point around a unit circle centered at the origin in
the
xy
-plane results in the point . Thus, the value of is the x-coordinate of the point , which is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of the cosine of a multiple of , not
.
Choice D is incorrect. This is the value of , not .
Page 11
Question ID: 800e71b8
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
Points , , and lie on the circle shown. On this circle, minor arc
has a length of centimeters and major arc has a length of
centimeters. What is the circumference, in centimeters, of the circle shown?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. Since the endpoints of minor arc and major arc are the same, and the arcs together form a full circle, the sum
of the lengths of these two arcs is equal to the circumference of the circle. It's given that the length of minor arc is centimeters and the
length of major arc is centimeters. Therefore, the circumference of the circle, in centimeters, is , or .
Choice A is incorrect. This is the length, in centimeters, of minor arc
, not the circumference, in centimeters, of the circle.
Choice B is incorrect. This is the difference of the lengths of major arc and minor arc
, in centimeters.
Choice C is incorrect. This is the length, in centimeters, of major arc
, not the circumference, in centimeters, of the circle.
Page 12
Question ID: 9e44284b
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
In thexy-plane, the graph of is a circle. What is the radius of the circle?
Answer
A. 5
B. 6.5
C.
D.
Correct Answer: A
Rationale
Choice A is correct. One way to find the radius of the circle is to rewrite the given equation in standard form, , where
is the center of the circle and the radius of the circle is r. To do this, divide the original equation, , by 2 to
make the leading coefficients of and each equal to 1:
. Then complete the square to put the equation in
standard form. To do so, first rewrite as .
Second, add 2.25 and 0.25 to both sides of the equation: . Since
, , and , it follows that . Therefore,
the radius of the circle is 5.
Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of
the equation of a circle in the xy-plane.
Page 13
Question ID: fc8aa563
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
What is the center of the circle in the xy-plane defined by the equation ?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. The equation of a circle in the xy-plane can be written as , where the center of the circle is
and the radius of the circle is . It's given that the circle in the xy-plane is defined by the equation . This equationcan
be written as . For this equation, it follows that and . Therefore, the center of the circle in the xy-
plane defined by the given equation is .
Choice A is incorrect. This is the center of the circle in the xy-plane that is defined by the equation , not
.
Choice B is incorrect. This is the center of the circle in the xy-plane that is defined by the equation , not
.
Choice D is incorrect. This is the center of the circle in the xy-plane that is defined by the equation , not
.
Page 14
Question ID: 2855cb58
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
A circle in the xy-plane has its center at and has a radius of . Which equation represents this circle?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. The equation of a circle in the xy-plane can be written as , where the center of the circle is
and the radius of the circle is . It’s given that this circle has a center at and a radius of . Substituting for , for , and for
in
yields , or . Therefore, the equation that
represents this circle is .
Choice A is incorrect. This equation represents a circle with radius
, not.
Choice C is incorrect. This equation represents a circle with radius
, not.
Choice D is incorrect. This equation represents a circle with radius
, not.
Page 15
Question ID: 74d8b897
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
An angle has a measure of radians. What is the measure of the angle in degrees?
Correct Answer: 81
Rationale
The correct answer is . The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by . Multiplying the
given angle measure, radians, by
yields , which is equivalent to degrees.
Page 16
Question ID: ee540927
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
In the xy-plane, the graph of the given equation is a circle. What are the coordinates
of the center of the circle?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. It’s given that in the xy-plane, the graph of is a circle. The equation of a circle in the xy-plane can be
written as
, where the coordinates of the center of the circle are and the radius of the circle is . By completing
the square, the equation
can be rewritten as , or
. This equation is equivalent to , or . Therefore, is
and is , and the
coordinates
of the center of the circle are .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Page 17
Question ID: a0cacec1
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
An angle has a measure of radians. What is the measure of the angle, in degrees?
Correct Answer: 192
Rationale
The correct answer is . The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by . Multiplying
the given angle measure, , by
yields , which simplifies to degrees.
Page 18
Question ID: 1b2b20b9
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
Circle A shown is defined by the equation
. Circle B (not shown) has the same radius but is translated units to the right. If
the equation of circle B is , where , , and are constants, what is the value of ?
Correct Answer: 28
Rationale
The correct answer is . The equation of a circle in the xy-plane can be written as , where the center of the circle is
and the radius of the circle is . It’s given that circle A is defined by the equation , which can be written as
. It follows that and the radius of circle A is . It’s also given that circle B has the same radius as circle A. If the
equation of circle B is , then . Substituting for in this equation yields , or . It follows
that the value of
is , or .
Page 19
Question ID: 23c5fcce
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Easy
Question
The circle above with center O has a circumference of 36. What is the length of minor arc ?
Answer
A. 9
B. 12
C. 18
D. 36
Correct Answer: A
Rationale
Choice A is correct. A circle has 360 degrees of arc. In the circle shown, O is the center of the circle and is a central angle of the circle.
From the figure, the two diameters that meet to form are perpendicular, so the measure of is . Therefore, the length of
minor arc is of the circumference of the circle. Since the circumference of the circle is 36, the length of minor arc is
.
Choices B, C, and D are incorrect. The perpendicular diameters divide the circumference of the circle into four equal arcs; therefore, minor arc
is of the circumference. However, the lengths in choices B and C are, respectively, and the circumference of the circle, and the
length in choice D is the length of the entire circumference. None of these lengths is the circumference.
Page 20
Question ID: 2d521ca9
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
The measure of angle is . What is the measure, in radians, of angle ?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. The measure of an angle, in radians, can be found by multiplying its measure, in degrees, by . It's given that the measure
of angle is . It follows thatthe measure, in radians, of angle is , or .
Choice A is incorrect. This is the measure, in radians, of an angle whose measure is , not .
Choice C is incorrect. This is the measure, in radians, of an angle whose measure is , not .
Choice D is incorrect. This is the measure, in radians, of an angle whose measure is , not .
Page 21
Question ID: ca2235f6
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
A circle has center , and points and lie on the circle. The measure of arc is and the length of arc is inches. What is the
circumference, in inches, of the circle?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. It’s given that the measure of arc is and the length of arc is . The arc measure of the full circle is .
If represents the circumference, in inches, of the circle, it follows that . This equation is equivalent to , or .
Multiplying both sides of this equation by yields , or . Therefore, the circumference of the circle is .
Choice A is incorrect. This is the length of arc .
Choice B is incorrect and may result from multiplying the length of arc by .
Choice C is incorrect and may result from squaring the length of arc .
Page 22
Question ID: 856372ca
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
In the xy-plane, a circle with radius 5 has center . Which of the following is an equation of the circle?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. An equation of a circle is , where the center of the circle is and the radius is r. It’s given that
the center of this circle is and the radius is 5. Substituting these values into the equation gives , or
.
Choice A is incorrect. This is an equation of a circle that has center . Choice C is incorrect. This is an equation of a circle that has center
and radius . Choice D is incorrect. This is an equation of a circle that has radius .
Page 23
Question ID: 981275d2
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
In the xy-plane, the graph of the equation above is a circle. Point P is on the circle and has coordinates . If is a diameter of the
circle, what are the coordinates of point Q ?
Answer
A.
B.
C.
D.
Correct Answer: A
Rationale
Choice A is correct. The standard form for the equation of a circle is , where are the coordinates of the center
and r is the length of the radius. According to the given equation, the center of the circle is . Let represent the coordinates of
point Q. Since point P and point Q are the endpoints of a diameter of the circle, the center lies on the diameter,
halfway between P and Q. Therefore, the following relationships hold: and . Solving the equations for
and , respectively, yields and . Therefore, the coordinates of point Q are .
Alternate approach: Since point P on the circle and the center of the circle have the same y-coordinate, it follows that the
radius of the circle is . In addition, the opposite end of the diameter must have the same y-coordinate as P and be 4 units away
from the center. Hence, the coordinates of point Q must be .
Choices B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter . If either of
these points were point Q, then would not be the diameter of the circle. Choice C is incorrect because is the center of the circle
and does not lie on the circle.
Page 24
Question ID: 89661424
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
A circle in the xy-plane has its center at and has a radius of . An equation of this circle is , where , , and
are constants. What is the value of ?
Correct Answer: -52
Rationale
The correct answer is . The equation of a circle in the xy-plane with its center at
and a radius of can be written in the form
. It's given that a circle in the xy-plane has its center at
and has a radius of . Substituting for , for , and for
in the equation
yields , or . It's also given that an
equation of this circle is , where , , and are constants. Therefore,
can be rewritten
in the form . The equation , or , can be
rewritten as
. Combining like terms on the left-hand side of this equation yields
. Subtracting from both sides of this equation yields , which is equivalent to
. This equation is in the form . Therefore, the value of is .
Page 25
Question ID: d03e29f1
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Easy
Question
The graph of the given equation in the xy-plane is a circle. What is the length of the radius of this circle?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. The equation of a circle in the xy-plane can be written as , where the center of the circle is
and the radius of the circle is . The graph of the given equation,
, is a circle in the xy-plane. This equation can be
written as
, where , , and . Therefore, the radius of this circle is .
Choice A is incorrect. This is the y-coordinate of the center, not the radius, of the circle defined by the given equation.
Choice B is incorrect. This is the x-coordinate of the center, not the radius, of the circle defined by the given equation.
Choice D is incorrect. This is the value of the radius squared, not the radius, of the circle defined by the given equation.
Page 26
Question ID: 95ba2d09
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
In the xy-plane above, points P, Q, R, and T lie on the circle with center O. The degree measures of angles and are each 30°. What is
the radian measure of angle ?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. Because points T, O, and P all lie on the x-axis, they form a line. Since the angles on a line add up to , and it’s given that
angles POQ and ROT each measure , it follows that the measure of angle
QOR
is . Since the arc of a
complete circle is or radians, a proportion can be set up to convert the measure of angle QOR from degrees to radians:
, where x is the radian measure of angle QOR. Multiplying each side of the proportion by gives
. Solving for x gives , or .
Choice A is incorrect and may result from subtracting only angle POQ from to get a value of and then finding the radian measure
equivalent to that value. Choice B is incorrect and may result from a calculation error. Choice D is incorrect and may result from calculating the
sum of the angle measures, in radians, of angles POQ and ROT.
Page 27
Question ID: fb58c0db
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
Points A and B lie on a circle with radius 1, and arc has length . What fraction of the circumference of the circle is the length of arc
?
Rationale
The correct answer is . The circumference, C, of a circle is , where r is the length of the radius of the circle. For the given circle with
a radius of 1, the circumference is , or . To find what fraction of the circumference the length of arc is, divide the
length of the arc by the circumference, which gives . This division can be represented by . Note that 1/6, .1666,
.1667, 0.166, and 0.167 are examples of ways to enter a correct answer.
Page 28
Question ID: acd30391
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Hard
Question
A circle in the xy-plane has equation . Which of the following points does NOT lie in the interior of the circle?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. The circle with equation has center and radius 5. For a point to be inside of the circle,
the distance from that point to the center must be less than the radius, 5. The distance between and is
, which is greater than 5. Therefore, does NOT lie in the interior of the circle.
Choice A is incorrect. The distance between and is , which is less
than 5, and therefore lies in the interior of the circle. Choice B is incorrect because it is the center of the circle. Choice C is incorrect
because the distance between and is , which is less than 5, and therefore
in the interior of the circle.
Page 29
Question ID: 82c8325f
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
A circle in thexy
-plane has its center at
and the point lies on the circle. Which equation represents this circle?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. A circle in the xy-plane can be represented by an equation of the form
, where
is the center
of the circle and
is the length of a radius of the circle. It's given that the circle has its center at
. Therefore, and .
Substituting
for and for in the equation
yields
, or
. It's also given that the point
lies on the circle. Substituting for and for in the equation yields
, or , which is equivalent to
, or
. Substituting for in the equation
yields
. Thus, the equation
represents the circle.
Choice A is incorrect. The circle represented by this equation has its center at
, not
, and the point
doesn't lie on the
circle.
Choice B is incorrect. The point
doesn't lie on the circle represented by this equation.
Choice C is incorrect. The circle represented by this equation has its center at , not , and the point
doesn't lie on the
circle.
Page 30
Question ID: b96ff36e
Assessment Test Domain Skill Difficulty
SAT Math Geometry and
Trigonometry
Circles Medium
Question
In the xy-plane, the graph of the equation is a circle. The point , where is a constant, lies on this circle. What is
the value of
?
Correct Answer: 5
Rationale
The correct answer is . It's given that in the xy-plane, the graph of the equation
is a circle. It’s also given that the
point , where is a constant, lies on this circle. It follows that the ordered pair
makes the equation
true.
Substituting for and for in this equation yields , or . Subtracting from each side of this
equation yields . It follows thatthe value of is .
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