Analytic Geometry

ANALYTIC GEOMETRY

Analytic Geometry

One of the great achievements in mathematics that occurred in the early 1600s was the merging of two branches of mathematics— mathematics —algebra and geometry geometry —into a single, unified body of knowledge, now known as analytic geometry .

Analytic Geometry

One of the great achievements in mathematics that occurred in the early 1600s was the merging of two branches of mathematics— mathematics —algebra and geometry geometry —into a single, unified body of knowledge, now known as analytic geometry .

Analytic Geometry

Credit for this great accomplishment goes to the French mathematician Rene Descartes (1596-1650). The same method was also independently discovered by another famous French mathematician Pierre de Fermat (16011665). Descartes’ s unifying concept became not only the basis for analytic geometry but had been most fruitful in the development of calculus.

POINTS IN A PLANE

Points in a Plane

Objectives 1. Describe the basic concepts of the Cartesian coordinate system which include the coordinate axes, ordered pairs, abscissa, ordinate, quadrants, and origin. 2. Plot ordered pairs of real numbers in the rectangular coordinate system. 3. Determine the coordinates of given points in the xy-plane.

Points in a Plane

4. Find the increment in x, denoted by ∆x, and increment in y, denoted by ∆y, as a particle moves from one point to another in the xy-plane. 5. Compute the distance between two points. 6. Classify whether a triangle formed by three given points is a right triangle or an isosceles triangle.

Points in a Plane

7. Locate the midpoint of a line segment joining two points using analytical and graphical methods. 8. Sketch the graph of a given equation and determine its: a. intercepts b. symmetry c. asymptotes

The Cartesian Coordinate System


It was the French mathematician and philosopher Rene Descartes who is credited with the idea of unifying algebra with geometry. This field of study is called analytic geometry . Descartes associated every point in the plane with a pair of real numbers resulting in the xycoordinate system. This system is also referred to as the Cartesian coordinate system in his honor.


The xy-coordinate system is based on two perpendicular number lines. Any point on a number line is associated with a single real number which we call its coordinate . The origin of a number line is its zero point . The two real number lines intersect at their zero points and usually have the same unit of length drawn in the figure.


These number lines are called the coordinate axes . The horizontal axis is called the x-axis and the vertical axis is called the y-axis . The point where they intersect is called the origin .


The coordinate axes divide the whole plane into four regions called quadrants that are labeled from the upper right corner, counterclockwise.


For the horizontal axis, the zero point is at the origin. Thus, all points to the right of the origin on the x-axis are positive; all points to the left of the origin are negative. On the y-axis, all points extending upward from the origin are positive; all points downward from the origin are negative.


All points are symmetric about the origin. This means that +a on the x-axis which lies a units to the right from the origin has a mirror image of –a which is a units to the left of the origin. On the y-axis, +b above the origin has a mirror image of –b below the origin. Because the references in this system are the two-coordinate axes—the x-axis and the y-axis— this system is called the xy-coordinate system.


Every point in the Cartesian coordinate system is associated with a pair of real numbers (a, b). From any point P, we draw perpendicular lines to the coordinate axes. Then we assign a pair of real numbers (a, b) to the point.


The number a corresponds to the point on the x-axis and is called the x-coordinate of P. It is the distance from the y-axis to the point P. The number b corresponds to the point on the y-axis and is called the y-coordinate of P. It is the distance from the x-axis to the point P.


The pair of real numbers (a, b) is called the coordinate pair of point P. In symbols, we write P(a, b) to designate that point P has the coordinate pair (a, b).


It is important to note that the coordinate pair is an ordered pair which implies that the order of the real numbers is important. The xcoordinate, also called the abscissa , is the first number a, while the y-coordinate, also called as the ordinate , is the second number b.


We call the set of all points determined by the ordered pair (a, b) as the xy-plane or the Cartesian plane . There is a one-to-one correspondence between the points in the xyplane and the set of all ordered pairs of real numbers. This means that each point in the xyplane corresponds to an ordered pair of real numbers. Also, each ordered pair of real numbers represent a point in the xy-plane.

Points in the Four Quadrants


The signs of the coordinates determine the quadrant where the point lies. In the first quadrant, the two coordinates are both positive or greater than zero. In the second quadrant, the x-coordinate is negative (x ‹ 0), while the ycoordinate is positive (y › 0). Both coordinates are negative in the third quadrant. In the fourth quadrant, the abscissa is positive (x › 0), while the ordinate is negative (y ‹ 0).


Signs of the Coordinates in the Four Quadrants First Quadrant (Q-I)

x>0

y>0

Second Quadrant (Q-II)

x<0

y>0

Third Quadrant (Q-III)

x<0

y<0

Fourth Quadrant (Q-IV)

x>0

y<0


Example 1.1 Tell the quadrant where each of the following ordered pairs are found. a. (-1, 2)

Q II

b. (3, 2)

QI

c. (2, -5)

Q IV

d. (-1, -2)

Q III


In representing points in the plane, we may use capital letters such as P, Q, A, or B. Subscripts are also used in order to differentiate one point from another. For example, two distinct points may be represented by P1 and P2.

Locating a Point in the Plane


How do we locate a point in the rectangular coordinate system? In locating the point (a, b) in the coordinate system, the abscissa tells whether we go right or left of the origin, while the ordinate tells whether we go up or down do wn the x-axis.


Example 1.2 Locate the points A(4, -3) and B(-3, 4) in the Cartesian coordinate system.

Solution: We always start from the origin. For point A, the abscissa is 4 and the ordinate is -3.



Solution: So the point is located in Q IV. We count 4 units to the right from the origin and then go 3 units down.



Solution: For B(-3, 4), we count 3 units to the left from the origin and go 4 units upward. This point is in Q II.

Points on the Coordinate Axes


When either the x-coordinate or ycoordinate is zero, the point is either on the xaxis or on the y-axis. It is on the x-axis if y = 0, so the coordinates are either (+x, 0) or (-x, 0). It is on the y-axis if x = 0, so the coordinates are either (0, +y) or (0, -y).


Example 1.3 Locate the points C(0, -5) and D(6, 0) in the xyplane.

Solution: For point C the xcoordinate is zero, so the point is on the y-axis, 5 units below the origin.


Example 1.3 Locate the points C(0, -5) and D(6, 0) in the xyplane.

Solution: The y-coordinate of point D is zero, so the point is on the x-axis, 6 units to the right of the origin.

Finding the Coordinates of a Point


How do we determine the coordinates of a point? We simply draw perpendicular lines from the point to the two coordinate axes and find the coordinates.


Example 1.4 Determine the coordinates of points P 1 and P2.

Solution: P1 is on the x-axis so its ordinate is 0. It is 4 units to the right of the origin, hence its coordinates are (4, 0).

P1


Example 1.4 Determine the coordinates of points P 1 and P2.

Solution: We draw perpendicular lines from P2 to the coordinate axes and indicate that its xcoordinate is -3 and its y-coordinate is -4. So, P2 is (-3, -4).

P2

P1

Increments in x and y


When a particle moves from one point to another in the Cartesian plane, there are changes in its coordinates. These net changes, called the increments of the coordinates , are determined by subtracting the coordinates of the starting point (initial point) from the corresponding coordinates of the point where the particle stops (final point).



In general, when a particle moves from P1(x1, y1) to P2(x2, y2), the increment in x is

∆x = x2 – x1

and the increment in y is

∆y = y2 – y1

where ∆x (read as “delta x”) stands for the increment in x and ∆y (read as “delta y”) stand for the increment in y. Note that ∆ is the Greek letter delta and stands for “difference .”


Example 1.5 Find the increments in the x and y coordinates as a particle moves from M(3, -2) to N(-1, 5). Solution: The initial point is M(3, -2) and the final point is N(-1, 5).

(x1, y1) = (3, -2) and (x2, y2) = (-1, 5)


Example 1.5 Find the increments in the x and y coordinates as a particle moves from M(3, -2) to N(-1, 5). Solution: The increment in x, ∆x = x2 – x1 = (-1) – (3) = -4 The increment in y, ∆y = y2 – y1 = 5 – (-2) = 7


Example 1.6 Find the increments in the x and y coordinates as a particle moves from M(5, -4) to N(5, 6). Solution: x1 = 5, y1 = -4 x2 = -1, y2 = 5 The increment in x, ∆x = 5 – 5 =0 The increment in y, ∆y = 6 – (-4) = 10


We see that the increments in the coordinates may be positive , negative , or zero . The sign tells the direction of the movement of the particle. That is, a positive increment is directed to the right while a negative negativ e increment is directed to the left. A positive increment in y is directed upward, while a negative increment in y is directed downward. A zero increment means no change with respect to the coordinate. This implies that the movement is perpendicular to that coordinate axis.

The Distance Between Two Points


The distance between the points M(x 1, 0) and N(x2, 0) on the x-axis is the absolute value v alue of the increment in x (∆x) (∆x) or

MN = |∆x| = |x2 – x1|

where MN is the symbol for the distance between M and N. Note that since the distance between any two points on the x-axis is the absolute value of the increment increment in x, then then it is always nonnegative.


The distance between the points X(0, y 1) and Y(0, y2) on the y-axis is the absolute value of the increment in y (∆y) or

XY = |∆y| = |y2 – y1|

Again, since we take the absolute value, then the distance is positive.


Example 1.7 Determine the distance between the points P 1(5, 8) and P2(-3, 8).

Solution: x1 = 5, y1 = 8 x2 = -3, y2 = 8 Since the two points have the same ycoordinate and P1P2 is parallel to the x-axis, we use P1P2 = |∆x|.


Example 1.7 Determine the distance between the points P 1(5, 8) and P2(-3, 8).

Solution: P1P2 = |∆x| = |x2 – x1| = |-3 – 5| = |-8| =8


Example 1.8 Find the distance between points P 1(5, -4) and P2(5, 6).

Solution: x1 = 5, y1 = -4 x2 = 5, y2 = 6 P1P2 = |∆y| = |y2 – y1| = |6 – (-4)| = |10| = 10


Suppose we now have two points, P(x 1, y1) and Q(x2 , y2). These two points do not necessarily have the same coordinates.


We draw a line through P parallel to the yaxis. Every point on this line has an x-coordinate of x1. Also, we draw a line through Q parallel to the x-axis. Every point on this line has a ycoordinate of y2. Let R be the point of intersection of the lines drawn. This point has the coordinates R(x1, y2).


Since PR is parallel to the y-axis, then

PR = |∆y| = |y2 – y1| and since QR is parallel to the x-axis, then

QR = |∆x| = |x2 – x1|


At point R, the angle formed by the intersecting lines is a right angle. Hence, it is a right triangle with hypotenuse PQ. Then, to find the distance between P and Q, we can apply the Pythagorean theorem.

(PQ)2 = (x2 – x1)2 + (y2 – y1)2


Example 1.9 Find the distance between P(2, -2) and Q(6, 1).

Solution: Let (x1, y1) = (2, -2) (x2, y2) = (6, 1) PQ =√(6-2)2 + [1–(-2)]2 =√42 + 32 =√25 =5

The Midpoint Formula


The midpoint between two points is defined as the point halfway or in the middle of the line segment that joins the two points. It is equidistant from the two points. To find the midpoint of a line, we use the midpoint formula,

M = x1 + x2 , y1 + y2

(

2

2

)


Example 1.10 Determine the midpoint of the line segment that joins P(-5, 2) and Q(3, 4).

Solution: There are two ways of solving this problem. The first method is the analytical method. The other method is the graphical method which involves plotting the points in the xy-plane and actually measuring the location of the midpoint and reading its coordinates.



Solution: a. Analytical Method With (x1, y1) = (-5, 2) and (x2, y2) = (3, 4), the apply the midpoint formula and get the xcoordinate of the midpoint as Mx = x1 + x2 = -5 + 3 = -2 = -1 2 2 2 My = y1 + y2 = 2 + 4 = 6 = 3 2 2 2



Solution: a. Analytical Method Therefore, the midpoint of PQ is M(-1, 3).



Solution: b. Graphical Method Points P(-5, 2) and Q(3, 4) are plotted in the figure. Using a ruler, the midpoint M is located where its coordinates are (-1, 3).

The Graph of an Equation


In this section, the discussion will be limited to the steps followed in sketching the graph of an equation. Some characteristics of the graph of an equation like the intercepts, its symmetry, and asymptotes are also discussed. The more complicated concepts and properties are dealt with in more advanced courses.


An equation is a statement that indicates the equality of two mathematical expressions involving variable(s). The open sentence 3x – 2 = 1 is an equation in one variable x, while the open sentence 2x + 5y = 1 is an equation in two variables x and y. When x = 1 is substituted in the first equation, we have 3x – 2 = 1 3(1) – 2 = 1 3–2=1 1 = 1 The sentence is true. The solution set for the equation 3x – 2 = { 1 }.


For the second equation, let us substitute x = -2 and y = 1. We obtain the following: 2x + 5y = 1 2(-2) + 5(1) = 1 -4 + 5 = 1 The sentence is true. 1=1 Therefore, the ordered pair (-2, 1) is a solution to the equation 2x + 5y = 1.


The set of all ordered pairs (x, y) that will make the equation in two variables x and y true is called the solution set of the equation. When the solution set of an equation is plotted in the rectangular coordinate system, it describes the graph of the equation. The graph of the equation is the set of points whose coordinates (x, y) satisfy the equation.


The graph of an equation may be a straight line or a curve. Every point on the line or the curve satisfy the equation and, conversely, every ordered pair (x, y) that satisfies the equation lies on the line or curve.


Sketching the Graph of an Equation How do we sketch the graph of a given equation in one or two variables? Basically, there are three steps to follow: 1. Choose some values for either of the variables and substitute these values in the equation to solve for the other variable.


Sketching the Graph of an Equation 2. Set up the table of values of x and y and obtain the corresponding ordered pairs (x, y). 3. Plot the points representing the ordered pairs in the Cartesian plane and join the points by a straight line or a smooth curve.


Example 1.11 Sketch the graph of the equation x + y = 5. Choose three points and connect them. Solution: Step 1. Assign values for x and y. Values of x

Values of y, where y = 5 – x

x1 = 0

y1 = 5 – 0 = 5

x2 = 1

y2 = 5 – 1 = 4

x3 = 2

y3 = 5 – 2 = 3


Example 1.11 Sketch the graph of the equation x + y = 5. Choose three points and connect them. Solution: Step 2. Set up the table of values. x

y

(x, y)

0

5

(0, 5)

1

4

(1, 4)

2

3

(2, 3)


Example 1.11 Sketch the graph of the equation x + y = 5. Choose three points and connect them. Solution: Step 3. Plot the points and connect them.


Intercepts of the Graph The points where the graph intersects the coordinate axes (both the x-axis and the y-axis) are considered important points. The x-intercept is the x-coordinate of the point where the graph intersects the x-axis. It is the value of the variable x when y = 0. When the x-intercept is equal to a, the point has the coordinates (a, 0).


Intercepts of the Graph The y-intercept is the y-coordinate of the point of intersection between the graph and the yaxis. Thus, a graph whose y-intercept is b intersects the y-axis at the point (0, b).


The graph of an equation may have several intercepts or none at all. A graph has no xintercept when it does not intersect either one of the coordinate axes. If the curve crosses only the y-axis, it has no x-intercept. If it intersects only the x-axis, it has no y-intercept.


1. In this figure, there is no intercept . The graph does not cross the x-axis nor the y-axis.


2. In this figure, there is one intercept, A(1, 0). The graph intersects only the x-axis and there is no y-intercept.


3. In this figure, there are two intercepts. The x-intercept is 2, while the yintercept is -3.


4. In this figure, there are three intercepts: two x-intercepts and one y-intercept. The two xintercepts are 2 and -2, while the y-intercept is -1.


5. In this figure, there are four intercepts: two xintercepts and two y-intercepts. The two xintercepts are 3 and -3, while the two y-intercepts are 5 and -5.


Symmetry About an Axis A curve has symmetry about an axis if there is a reflection or a mirror image of the curve on the other side of the axis of symmetry. This means that if the xy-plane is folded along the axis of symmetry , the graph on one side of the axis of symmetry will coincide with the graph on the other side. The axis of symmetry may be the x-axis, the y-axis, or any line in the plane.


1. In this figure, the curve has symmetry or is symmetric about the y-axis. The curve on the left side of the y-axis is the mirror image of the curve on the right side. Note that if the point (a, b) is on the graph, then so is (-a, b).


2. In this figure, the curve is symmetric about the x-axis. The curve above the xaxis has a refection below it. The axis of symmetry is the xaxis. We note also that if the point (a, b) is on the graph, then so is (a, -b).


3. Sometimes the axis of symmetry is neither of the coordinate axes. In this figure, the curve is symmetric about the line defined by the equation x + y = 1.


Symmetry About the Origin A graph may also be symmetric about the origin as presented in the figure. The graph at the first quadrant has a reflection at the third quadrant. Note that if (a, b) is a point of the graph, then so is (-a, -b).


Asymptote of the Curve In the figure, observe that the curve gets closer and closer to the line x = 2, but the curve does not cross the line. There are graphs of equations that approach a line but do not intersect it.


Asymptote of the Curve This line that the graph approaches but does not touch is called an asymptote. The asymptote may be the x-axis, the y-axis, or any line in the plane.

Important Terms Abscissa Asymptote Cartesian coordinate system Cartesian plane Coordinate axes Distance between two points Graph of an equation Increment Intercept Midpoint

Important Terms Ordered pair Ordinate Origin Quadrant Rectangular coordinate system Symmetry x-axis xy-plane y-axis

Summary o

o

Analytic geometry describes geometric figures by means of algebraic equations. The Cartesian coordinate system defines points, lines, and geometric figures in the plane with respect to two coordinate axes– the x-axis and the y-axis. Each point in the plane is in one-to-one correspondence with an ordered pair of real numbers.

Summary

o

o

To plot the ordered pair (a, b) in the Cartesian plane, the first number a is plotted along the x-axis, while the second number b is plotted along the y-axis. As a particle moves from point P(x 1, y1) to Q(x2, y2), it undergoes net changes in x called the increment in x. ∆x = x2 – x1 and net changes in y called the increment in y, ∆y = y2 – y1

Summary

o

The distance PQ between two points P(x1, y1) and Q(x2, y2) in the Cartesian coordinate system is equal to

PQ = √ (x2 – x1)2 + (y2 – y1)2 o

The midpoint of the line segment joining two points in the Cartesian coordinate system has the coordinates

M x1 + x2 , y1 + y2

(

2

2

)

Summary

o

o

The graph of an equation is the set of points whose coordinates satisfy the equation. To graph a given equation, we construct a table of values of ordered pairs that are solutions to the equation, plot the points , and then connect these points by either a straight line or a smooth curve. The intercepts are points of intersection of a graph and the x-axis or the y-axis.

Analytic Geometry

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