MATH I Reviewer

LICENSURE EXAMINATION FOR TEACHERS (LET)

LICENSURE EXAMINATION FOR TEACHERS (LET)

Refresher Course

Content Area: MATHEMATICS

Focus: ANALYTIC GEOMETRYPrepared by: Daisy de Borja-Marcelino

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 PointsA. The distance between two points (x1,y1) and (x2,y2) is given by .

Example: Given the points A(2,1) and B(5,4). Determine the length AB.

Solution: AB = or or .Exercises: For 1-2, use the following condition: Two insects L and O are initially at a point (-1,3) on a Cartesian plane.

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 iii7. What is the distance between (-5,-8) and (10,0)?

A) 17

B) 13

C) 23

D) -0.5B. Slope of a line

a) The slope of the non-vertical line containing A(x1,y1) and B(x2,y2) is or .

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: where (x1, y1) is any point on the line.

Two point form: where (x1, y1) and (x2, y2) are any two points on the line.

Intercept form: 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

uThe slopes of lines t, s, u are all negative. Of the three slopes, t is the highest, while u has the lowest (because the values are negative.)

Exercises1. What is the slope of ?

A)1.25

B) -1.25

C) 0.8

D) -0.82. 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 ?

A)

B) 2.25

C) -

D) - 2.25E. 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 .

If a point P divides in the ratio so that , then the coordinates of P(x,y) can be obtained using the formula and .

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

.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. Perpendicularc. 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

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.

b.

c.

d.

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 ,

the axis is the x-axis, the latus rectum is the line containing .

The graph of . 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 , then the equation is .

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 , then the equation is .

c) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is horizontal and , then the equation is .

d) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is vertical and , then the equation is .

3. Parts of an EllipseFor 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, ) and G(-c, ) and the other at H (c, ) and G(c, ).

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, 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, 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. and are the latus rectum, each with a length of .

The graph of .

The graph of .

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.

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 , then the equation is .

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 , then the equation is .

c) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is horizontal and , then the equation is .

d) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is vertical and , then the equation is

2. Parts of a hyperbolaFor 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, ) and G(-c, ) and the other at H (c, ) and H(c, ).

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 is the transverse axis.f) The line segment with endpoints (0,b) and (0,-b) where 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, is a latus rectum whose endpoints are G (-c, ) and G(-c, ) and has a length of .

3. The Asymptotes of a HyperbolaShown 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 are and . Similarly, the equations of the asymptotes associated with are and.

The graph of .

The graph of .

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. unitsb. 6 sq. units

c. unitsd. 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. unitsb. 10 sq. units

c. 5 unitsd. 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=3d. 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

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 = , 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 + 216. 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. units

b. 6 sq. units

c. 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. .

b. .c. .

d. .

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 from ?

a. 1 unit

b. 2 units

c. 3 units

d. 8 units

25. Which of the following is a focus of ?

a. (0,-4)

b. (-4,0)

c. (0,4)

d. (4,0)

26. What are the x-intercepts of ?

a. none

b.

c.

d.

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.

b.

c.

d.

For items 29-31, consider the graph on the right.

29. Which of the following is the equation of

the graph?

a.

b.

c.

d.

30. What are the x-intercepts of the graph?

a. none

b.

c.

d.

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

F(-c,0)

B(0,-b)

B(0,b)

O

y

x

3

-3

-2

-1

A

1

2

M

3

H

-1

0

x

1

2

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.

F(c,0)

V(-a,0)

V(a,0)

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

x

y

O

(-4,0)

(4,0)

(5,0)

EMBED Equation.3

(-5,0)

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

(0, -3)

(0, 3)

x

y

O

(12,1)

(2,-4)

(-8,1)

(2,6)

(-6,4)

(2,1)

(8,5)

(8,3)

y

y

y

x

D(a,0)

E(b,0)

0

O(0,b)

V

y

T

-3

-2

M

H

A

T

x

x

x

x

y

x

y

F(0,-6)

O

(0,3)

F(0,6)

(0,-3)

(-9,6)

(9,6)

EMBED Equation.3

EMBED Equation.3

x

y

F(-6,0)

O

(-3,0)

(3,0)

F(6,0)

(6,9)

(6,-9)

St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515 1

_1172347083.unknown

_1298488610.unknown

_1298490370.unknown

_1312839202.unknown

_1312839341.unknown

_1312917221.unknown

_1312917241.unknown

_1312839297.unknown

_1300275157.unknown

_1300275204.unknown

_1298490468.unknown

_1298490483.unknown

_1298490419.unknown

_1298489676.unknown

_1298489778.unknown

_1298489893.unknown

_1298489901.unknown

_1298489796.unknown

_1298489753.unknown

_1298488741.unknown

_1298488751.unknown

_1298488705.unknown

_1172689604.unknown

_1294430949.unknown

_1298321011.unknown

_1298488245.unknown

_1298488420.unknown

_1298488458.unknown

_1298488445.unknown

_1298488380.unknown

_1298488135.unknown

_1298488159.unknown

_1298321031.unknown

_1298320477.unknown

_1298320995.unknown

_1294860488.unknown

_1245600339.unknown

_1245600966.unknown

_1245601041.unknown

_1245610818.unknown

_1294430892.unknown

_1245610817.unknown

_1245600988.unknown

_1245600504.unknown

_1245600947.unknown

_1245600503.unknown

_1172730532.unknown

_1172732144.unknown

_1245600253.unknown

_1172732109.unknown

_1172729861.unknown

_1172730468.unknown

_1172689682.unknown

_1172367714.unknown

_1172368206.unknown

_1172369793.unknown

_1172514377.unknown

_1172514579.unknown

_1172514604.unknown

_1172514419.unknown

_1172370176.unknown

_1172368425.unknown

_1172367932.unknown

_1172367790.unknown

_1172366246.unknown

_1172366607.unknown

_1172366758.unknown

_1172366946.unknown

_1172366284.unknown

_1172366138.unknown

_1172297987.unknown

_1172298408.unknown

_1172346191.unknown

_1172346443.unknown

_1172346764.unknown

_1172346803.unknown

_1172346465.unknown

_1172346274.unknown

_1172298628.unknown

_1172298315.unknown

_1172298367.unknown

_1172298194.unknown

_1172258710.unknown

_1172259444.unknown

_1172297923.unknown

_1172258769.unknown

_1172258128.unknown

_1172258241.unknown

_1172257947.unknown

Documents

MATH I Reviewer