Math - analytic geometry

I N F I N I T H I N K Page 1
Refresher Course
Content Area: MATHEMATICS
Focus: ANALYTIC GEOMETRY
Competencies:
1. Solve problems involving coordinates of a point, midpoint of a line segment, and distance between
two points.
2. Determine the equation of the line relative to given conditions: slope of a line given its graph, or its
equation, or any two points on it.
3. Determine the equation of a non-vertical line given a point on it and the slope of a line, which is
either parallel or perpendicular to it.
4. Solve problems involving
a. the midpoint of a line segment, distance between two points, slopes of lines, distance
between a point and a line, and segment division.
b. a circle, parabola, ellipse, and hyperbola.
5. Determine the equations and graphs of a circle, parabola, ellipse and hyperbola.
I. The Cartesian Plane
Below is a diagram of a Cartesian plane or a rectangular coordinate system, or a coordinate plane.
An ordered pair of real numbers, called the coordinates of a point, locates a point in the Cartesian plane.
Each ordered pair corresponds to exactly one point in the Cartesian plane.
The following are the points in the figure on the right:
A(-6,3), B(-2,-3), C (4,-2), D(3,4), E(0,5), F(-3,0).
For numbers 1-2, use the following condition: Two insects M and T are initially at a point A(-4, -7) on
a Cartesian plane.
1. If M traveled 7 units to the right and 8 units downward, at what point is it now?
Solution: (-4+7, -7-8) or (-3,-15)
2. If T traveled 5 units to the left and 11 units downward, at what point is it now?
Solution: (-4-5, -7-11) or (-9, -18)
II. The Straight Line
A. Distance Between Two Points
A. The distance between two points (x1,y1) and (x2,y2) is given by 2
21
2
21 )()( yyxx −+− .
Example: Given the points A(2,1) and B(5,4). Determine the length AB.
Solution: AB = ( ) ( ) ( ) ( ) 1899394152
2222
=+=−+−=−+− or 29• or 23 .
Exercises: For 1-2, use the following condition: Two insects L and O are initially at a point (-1,3) on a
Cartesian plane.
The two axes separate the plane into four
regions called quadrants. Points can lie in one of
the four quadrants or on an axis. The points on
the x-axis to the right of the origin correspond to
positive numbers; while to the left of the origin
correspond to negative numbers. The points on
the y-axis above the origin correspond to positive
numbers; while below the origin correspond to
negative numbers.

1. If L traveled 5 units to the left and 4 units upward, at what point is it now?
A) (-6, 7) B) (4, 7) C) (-6, -1) D) (4, -7)
2. If O traveled 6 units to the right and 2 units upward, at what point is it now?
A) (7, 5) B) (5,5) C) (-7, 5) D) (-5, -5)
3. Two buses leave the same station at 9:00 p.m. One bus travels at the rate of 30 kph and the other travels at 40
kph. If they go on the same direction, how many km apart are the buses at 10:00 p.m.?
A) 70 km B) 10 km C) 140 km D) 50 km
4. Two buses leave the same station at 8:00 a.m. One bus travels at the rate of 30 kph and the other travels at 40
kph. If they go on opposite direction, how many km apart are the buses at 9:00 a.m.?
A) 70 km B) 10 km C) 140 km D) 50 km
5. Two buses leave the same station at 7:00 a.m. One bus travels north at the rate of 30 kph and the other travels
east at 40 kph. How many km apart are the buses at 8:00 a.m.?
A) 70 km B) 10 km C) 140 km D) 50 km
6. Which of the following is true about the quadrilateral with vertices A(0,0), B(-2,1), C(3,4) and D(5,3)?
i) AD and BC are equal
ii) BD and AC are equal
iii) AB and CD are equal
A) both i and iii B) ii only C) both ii and iii D) i, ii, and iii
7. What is the distance between (-5,-8) and (10,0)?
A) 17 B) 13 C) 23 D) -0.5
B. Slope of a line
a) The slope of the non-vertical line containing A(x1,y1) and B(x2,y2) is
21
21
xx
yy
m
−
−
= or
12
12
xx
yy
m
−
−
= .
b) The slope of the line parallel to the x-axis is 0.
c) The slope of the line parallel to the y-axis is undefined.
d) The slope of the line that leans to the right is positive.
e) The slope of the line that leans to the left is negative.
C. The Equation of the line
In general, a line has an equation of the form ax + by + c = 0 where a, b, c are real numbers and that a
and b are not both zero.
D. Different forms of the Equation of the line
• General form: ax + by + c = 0.
• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
• Point slope form: )( 11 xxmyy −=− where (x1, y1) is any point on the line.
• Two point form: )( 1
12
12
1 xx
xx
yy
yy −
−
−
=− where (x1, y1) and (x2, y2) are any two points on the line.
• Intercept form: 1=+
b
y
a
x
where a is the x-intercept and b is the y-intercept.
Reminders:
• A line that leans to the right has positive slope. The steeper the line, the higher the slope is.
p q r
The slopes of lines p, q, r are all positive. Of the three slopes, the slope of line p is the lowest; the slope of r is
the highest.

• A line that leans to the left has negative slope. The steeper the line, the lower the slope is.
t s u
The slopes of lines t, s, and u are all negative. Of the three slopes, t is the highest, while u has the lowest.
Exercises
1. What is the slope of 5 4 12 0x y- + = ?
A)1.25 B) -1.25 C) 0.8 D) -0.8
2. What is the slope of x = -9?
A) 4 B) 1 C) 0 D) undefined
3. What is the slope of y= 12?
A) 7 B) 1 C) 0 D) undefined
4. What is the slope of 1
4 9
x y
+ = ?
A) 4.0 B) 2.25 C) - 4.0 D) - 2.25
E. Parallel and Perpendicular lines
Given two non-vertical lines p and q so that p has slope m1 and q has slope m2.
• If p and q are parallel, then m1 = m2.
• If p and q are perpendicular to each other, then m1m2 = -1.
F. Segment division
Given segment AB with A(x1,y1) and B(x2,y2).
• The midpoint M of segment AB is )
2
,
2
( 2121 yyxx
M
++
.
• If a point P divides AB in the ratio
2
1
r
r
so that
2
1
r
r
PB
AP
= , then the coordinates of P(x,y) can be obtained
using the formula
21
1221
rr
xrxr
x
+
+
= and
21
1221
rr
yryr
y
+
+
= .
G. Distance of a point from a line
The distance of a point A(x1,y1) from the line Ax + By + C = 0 is given by
22
11
BA
CByAx
d
+
++
= .
Exercises
1. Write an equation in standard form for the line passing through (–2,3) and (3,4).
a. 5x – y = -13 b. x – 5y = 19 c. x – y = -5 d. x – 5y = –17
2. Write an equation in slope intercept form for the line with a slope of 3 and a y-intercept of 28.
a. y = –3x + 28 b. y = 0.5x + 28 c. y = 3x + 28 d. y = 3x + 21
3. Write the equation in standard form for a line with slope of 3 and a y-intercept of 7.
a. 3x – y = –7 b. 3x + y = 7 c. 3x + y = 7 d. –3x + y = –7
4. Which of the following best describes the graphs of 2x – 3y = 9 and 6x – 9y = 18?
a. Parallel b. Perpendicular c. Coinciding d. Intersecting
5. Write the standard equation of the line parallel to the graph of x – 2y – 6 = 0 and passing through (0,1).
a. x + 2y = –2 b. 2x – y = –2 c. x – 2y = –2 d. 2x + y = –2

x
y
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
6. Write the equation of the line perpendicular to the graph of x = 3 and passing through (4, –1).
a. x – 4 = 0 b. y + 1 = 0 c. x + 1 = 0 d. y – 4 = 0
7. For what value of d will the graph of 6x + dy = 6 be perpendicular to the graph 2x – 6y = 12?
a. 0.5 b. 2 c. 4 d. 5
III. Conic Section
A conic section or simply conic, is defined as the graph of a second-degree equation in x and y.
In terms of locus of points, a conic is defined as the path of a point, which moves so that its distance
from a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus of the
conic, the fixed line is called the directrix of the conic, and the constant ratio is called the eccentricity, usually
denoted by e.
If e < 1, the conic is an ellipse. (Note that a circle has e=0.)
If e = 1, the conic is a parabola.
If e > 1, the conic is hyperbola.
A. The Circle
1. A circle is the set of all points on a plane that are equidistant from a fixed point on the plane. The fixed
point is called the center, and the distance from the center to any point of the circle is called the radius.
2. Equation of a circle
a) general form: x2
+ y2
+ Dx + Ey + F = 0
b) center-radius form: (x – h)2
+ (y – k)2
= r2
where the center is at (h,k) and the radius is equal to r.
3. Line tangent to a circle
A line tangent to a circle touches the circle at exactly one point called the point of tangency. The tangent
line is perpendicular to the radius of the circle, at the point of tangency.
Exercises
For items 1-2, use the illustration on the right.
1. Which of the following does NOT lie on the circle?
a. (3,-1) b. (3,0)
c. (2,-1) d. (3,-2)
2. What is the equation of the graph?
a. ( ) 132
=−+ xy b. ( ) 13)1( 2
=−+− xy
c. ( ) 13)1( 2
=−++ xy d. ( ) 13)1( 2
=+++ xy
B. The Parabola
1. Definition. A parabola is the set of all points on a plane that are equidistant from a
fixed point and a fixed line of the plane. The fixed point is called the focus and the fixed line is the
directrix.
2. Equation and Graph of a Parabola
a) The equation of a parabola with vertex at the origin and focus at (a,0) is y2
= 4ax. The parabola
opens to the right if a > 0 and opens to the left if a < 0.
b) The equation of a parabola with vertex at the origin and focus at (0,a) is x2
= 4ay. The parabola
opens upward if a > 0 and opens downward if a < 0.
c) The equation of a parabola with vertex at (h , k) and focus at (h + a, k) is (y – k)2
= 4a(x – h).
The parabola opens to the right if a > 0 and opens to the left if a < 0.
d) The equation of a parabola with vertex at (h , k) and focus at (h, k + a) is (x – h)2
= 4a(y – k).

e) The parabola opens upward if a > 0 and opens downward if a < 0.
f) Standard form: (y – k)2
= 4a(x – h) or (x – h)2
= 4a(y – k)
g) General form: y2
+ Dx + Ey + F = 0, or x2
+ Dx + Ey + F = 0
3. Parts of a Parabola
a) The vertex is the point, midway between the focus and the directrix.
b) The axis of the parabola is the line containing the focus and perpendicular to the directrix. The
parabola is symmetric with respect to its axis.
c) The latus rectum is the chord drawn through the focus and parallel to the directrix (and therefore
perpendicular to the axis) of the parabola.
d) In the parabola y2
=4ax, the length of latus rectum is 4a, and the endpoints of the latus rectum are (a,
-2a) and (a, 2a).
In the figure at the right, the vertex of the parabola is the origin,
the focus is F(a,o), the directrix is the line containing 'LL ,
the axis is the x-axis, the latus rectum is the line containing 'CC .
The graph of yx
3
162
−= . The graph of (y-2)2
= 8 (x-3).
C. Ellipse
1. An ellipse is the set of all points P on a plane such that the sum of the distances of P from two fixed points
F’ and F on the plane is constant. Each fixed point is called focus (plural: foci).
2. Equation of an Ellipse
a) If the center is at the origin, the vertices are at (± a, 0), the foci are at (± c,0), the endpoints of the
minor axis are at (0, ± b) and 222
cab −= , then the equation is 12
2
2
2
=+
b
y
a
x
.
b) If the center is at the origin, the vertices are at (0, ± a), the foci are at (0, ± c), the endpoints of the
minor axis are at (± b, 0) and 222
2
2
2
=+
a
y
b
x
.

c) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is horizontal and
222
)()(
2
2
2
2
=
−
+
−
b
ky
a
hx
.
d) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is vertical and
222
)()(
2
2
2
2
=
−
+
−
b
hx
a
ky
.
3. Parts of an Ellipse
For the terms described below, refer to the ellipse
shown with center at O, vertices at V’(-a,0) and V(a,0),
foci at F’(-c,0) and F(c,0), endpoints of the minor axis
at B’(0,-b) and B(0,b), endpoints of one latus rectum
at G’ (-c,
a
b2
− ) and G(-c,
a
b2
) and the other at
H’ (c,
a
b2
− ) and G(c,
a
b2
).
a) The center of an ellipse is the midpoint of the segment joining the two foci. It is the intersection of
the axes of the ellipse. In the figure above, point O is the center.
b) The principal axis of the ellipse is the line containing the foci and intersecting the ellipse at its
vertices. The major axis is a segment of the principal axis whose endpoints are the vertices of the
ellipse. In the figure, VV ' is the major axis and has length of 2a units.
c) The minor axis is the perpendicular bisector of the major axis and whose endpoints are both on the
ellipse. In the figure, BB' is the minor axis and has length 2b units.
d) The latus rectum is the chord through a focus and perpendicular to the major axis. GG' and HH'
are the latus rectum, each with a length of
a
b2
2
.
The graph of 1
925
22
=+
yx
. The graph of 1
25
)1(
100
)2( 22
=
−
+
− yx
.
4. Kinds of Ellipses
a) Horizontal ellipse. An ellipse is horizontal if its principal axis is horizontal. The graphs above are both
horizontal ellipses.
b) Vertical ellipse. An ellipse is vertical if its principal axis is vertical.
D. The Hyperbola
1. A hyperbola is the set of points on a plane such that the difference of the distances of each point on the set
from two fixed points on the plane is constant. Each of the fixed points is called focus.
x
y
O
B(0,b
B’(0,-
F’(- F(c,0)V’(- V(a,0
),(
2
a
bc
),(
2
a
bc −
),(
2
a
bc−
),(
2
a
bc −−
x
y
O
(-4,0) (4,0) (5,0)
)
5
9(4,
(-5,0)
)
5
9(4,-
)
5
9(-4,
)
5
9(-4,-
(0, -3)
(0, 3)
x
y
O
(12,1)
(2,-4)
(-8,1)
(2,6)
(-6,4)
(2,1)
(8,5)
(8,3)

2. Equation of a hyperbola
a) If the center is at the origin, the vertices are at (± a, 0), the foci are at (± c,0), the endpoints of the minor
axis are at (0, ± b) and 222
acb −= , then the equation is 12
2
2
2
=−
b
y
a
x
.
b) If the center is at the origin, the vertices are at (0, ± a), the foci are at (0, ± c), the endpoints of the minor
axis are at (± b, 0) and 222
2
2
2
=−
b
x
a
y
.
c) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is horizontal and
222
)()(
2
2
2
2
=
−
−
−
b
ky
a
hx
.
d) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is vertical and
222
)()(
2
2
2
2
=
−
−
−
b
hx
a
ky
2. Parts of a hyperbola
For the terms described below, refer to the hyperbola shown which has its center at O, vertices at V’(-
a,0) and V(a,0), foci at F’(-c,0) and F(c,0) and endpoints of one latus rectum at G’ (-c,
a
b2
− ) and G(-c,
a
b2
) and the other at H’ (c,
a
b2
− ) and H(c,
a
b2
).
a) The hyperbola consists of two separate parts called branches.
b) The two fixed points are called foci. In the figure, the foci are at (± c,0).
c) The line containing the two foci is called the principal axis. In the
figure, the principal axis is the x-axis.
d) The vertices of a hyperbola are the points of intersection of the
hyperbola and the principal axis. In the figure, the vertices are at (± a,0).
e) The segment whose endpoints are the vertices is called the transverse axis. In the figure VV ' is the
transverse axis.
f) The line segment with endpoints (0,b) and (0,-b) where 222
acb −= is called the conjugate axis, and is a
perpendicular bisector of the transverse axis.
g) The intersection of the two axes is the center of the hyperbola .
h) The chord through a focus and perpendicular to the transverse axis is called a latus rectum. In the figure,
GG' is a latus rectum whose endpoints are G’ (-c,
a
b2
− ) and G(-c,
a
b2
) and has a length of
a
b2
2
.
3. The Asymptotes of a Hyperbola
Shown in the figure on the right is a hyperbola
with two lines as extended diagonals of the
rectangle shown.

These two diagonal lines are said to be the asymptotes of the curve, and are helpful in sketching the
graph of a hyperbola. The equations of the asymptotes associated with 12
2
2
2
=−
b
y
a
x
are x
a
b
y = and
x
a
b
y −= . Similarly, the equations of the asymptotes associated with 12
2
2
2
=−
b
x
a
y
are x
b
a
y = and x
b
a
y −= .
The graph of 1
279
22
=−
yx
. The graph of 1
279
22
=−
xy
.
PRACTICE EXERCISES
Directions: Choose the best answer from the choices given and write the corresponding letter of your choice.
1. Which of the following are the coordinates of A?
a. (1,2) b. (2,1) c. (-3,3) d. (2,-3)
2. What is the distance between points M and T?
a. 61 units b. 6 sq. units c. 51 units d. 8 units
3. Which of the following points has the coordinates (-3,-1)
a. M b. A c. T d. H
4. Which of the following is the area of the triangle formed with vertices M, A and H?
a. 5 sq. units b. 10 sq. units c. 5 units d. 10 units
5. Which of the following is the equation of the line containing points A and T?
a. y= 2 b. x=2 c. y+2x=3 d. y-2x+3=0
6. Suppose that an isosceles trapezoid is placed on the Cartesian plane as shown
On the right, which of the following should be the coordinates of vertex V?
a. (a,b) b. (b+a, 0) c. (b-a,b) d. (b+a,b)
7. The points (-11,3), (3,8) and (-8,-2) are vertices of what triangle?
a. Isosceles b. Scalene c. Equilateral d. Right
x
y
F’(-6,0) O(-3,0) (3,0) F(6,0)
(6,9)
(6,-9)
x
y
F’(0,-6)
O
(0,3)
F(0,6)
(0,-3)
(-9,6) (9,6)
03 =− xy
03 =+ xy
x
1
2
3
1 2 3-3 -2 -1
-3
-2
-1
M
A
T
H
y
x
D(a,0) E(b,0)
O(0,b) V
0
0

x
y
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
8. What is the area of the triangle in #7?
a. 40.5 sq units b. 41.8 sq units c. 42 sq units d. 46.8 sq units
9. Which of the following sets of points lie on a straight line?
a. (2,3), (-4,7), (5,8) b. (-2,1), (3,2), (6,3) c. (-1,-4), (2,5), (7,-2) d. (4,1), (5,-2), (6,-5)
10. If the point (9,2) divides the segment of the line from P1(6,8) to P2(x2,y2) in the ratio r =
7
3
, give the
coordinates of P2.
a. (16,–12) b. (–16, 15) c. (14,15) d. (12,–12)
11. Give the fourth vertex, at the third quadrant, of the parallelogram whose three vertices are (-1,-5), (2,1) and
(1,5).
a. (-3,-2) c. (-3,-4) c. (-4,-1) d. (-2,-1)
12. The line segment joining A(-2,-1) and B(3,3) is extended to C. If BC = 3AB, give the coordinates of C.
a. (17,12) b. (15,17) c. (18,15) d. (12,18)
13. The line L2 makes an angle of 600
with the L1. If the slope of L1 is 1, give the slope of L2.
a. (3 + 20.5
) b. (2 + 20.5
) c. –(2 + 30.5
) d. –(3 + 30.5
)
14. The angle from the line through (-4,5) and (3,m) to the line (-2,4) and (9,1) is 1350
. Give the value of m.
a.7 b. 8 c. 9 d. 10
15. Which equation represents a line perpendicular to the graph of 2x + y = 2?
a. y = -0.5x – 2 b. y = –2x + 2 c. y = 2x – 2 d. y = 0.5x + 2
16. Which of the following is the y – intercept of the graph 2x – 2y + 8 = 0?
a. -4 b. -2 c. 2 d. 4
17. Which of the following may be a graph of x – y = a where a is a positive real number?
a. b. c. d.
18. Write an equation in standard form for a line with a slope of –1 passing through (2,1).
a. x + y = –3 b. –x + y = 3 c. x + y = 3 d. x – y = –3
19. Which of the following are the coordinates of A?
a. (1,1) b. (1,-1)
c. (-1,1) d. (-1,-1)
20. What is the distance between points A and H?
a. 61 units b. 6 sq. units
c. 51 units d. 8 units
21. Which of the following points has the coordinates (-2,-2)?
a. M b. A
c. T d. H
22. Which of the following is the equation of the given graph?
a. ( ) 22
+= xy . b. ( ) 22
+−= xy . c. ( ) 22
−= xy . d. ( ) 22
−−= xy .
23. Which of the following is the equation of the line containing points M and T?
a. y= 2 b. x=2 c. y-2x-2=0 d. y+2x+2=0
24. What is the shortest distance of yx 82
= from 3=x ?
a. 1 unit b. 2 units c. 3 units d. 8 units
25. Which of the following is a focus of 1
412
22
=−
xy
?
y y y y
x x x x
A
HM
T

x
y
-30 -20 -10 0 10 20 30
-10
0
10
x
y
-30 -20 -10 0 10 20 30
-10
0
10
x
y
-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30
-15
-10
-5
0
5
10
15
x
y
-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30
-15
-10
-5
0
5
10
15
x
y
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
a. (0,-4) b. (-4,0) c. (0,4) d. (4,0)
26. What are the x-intercepts of 1
94
22
=+
yx
?
a. none b. 2± c. 3± d. 4±
27. Which of the following is a graph of a hyperbola?
a. b.
c. d.
28. Which of the following is an equation of an ellipse that has 10 as length of the major axis and has foci which
are 4 units away from the center?
a. 1
925
22
=+
xy
b. 1
169
22
=+
xy
c. 1
35
22
=+
xy
d. 1
2516
22
=+
xy
For items 29-31, consider the graph on the right.
29. Which of the following is the equation of
the graph?
a. 250025100 22
=+ xy
b. 250025100 22
=+ yx
c. 250025100 22
=− xy
d. 250025100 22
=− yx
30. What are the x-intercepts of the graph?
a. none b. 2±
c. 5± d. 10±
31. What kind of figure is shown on the graph?
a. circle b. ellipse c. hyperbola d. Parabola
32. Which of the following is the center of the graph
shown on the right?
a. (0,0) b. (0,10)
c. (10,0) c. (0,-10)
33. Which of the following is a focus of the graph
shown on the right?
a. (0,0) b. (0,10)
c. (0,5) c. (0,-10)
34. What is the area of the shaded region?
a. 4 units b. 4 square units
c. 16 units d. 16 square units
x
y
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2

Math - analytic geometry

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Math - analytic geometry