Analytic Geometry 1 -1 Cartesian Coordinate System - - PDF document

Analytic Geometry

ةيليلحتلا ةسدنھلا 1 -1 Cartesian Coordinate System يتراكيدلا تايثادحلئا ماظن

• The Cartesian coordinate system, or the rectangular coordinate system, is a

geometrical system that is used to determine the locations of points in a plane.

• Points are located with respect to a reference point called the origin which

is the intersection point of a horizontal line, known as x-axis, and a vertical line called y-axis.

• The x and y axes divide the Cartesian plane into four regions called

quadrants.

• Each point in the plane is defined by an ordered pair (x, y) of real numbers

called the coordinates of the point.

• An example of ordered pairs or coordinates is the point P below:

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1 - 2 The Distance Formula ةفاسملا داجيإ نوناق

• The distance d between two points A(x1, y1) and B(x2, y2) can be

found from the distance formula:

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Example 1: Find the distance between the points C(3, ‒ 4) and D( ‒ 13, ‒ 11).

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Solution:

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1 - 3 The Midpoint Formula فصتنملا ةطقن داجيإ نوناق

• The coordinates of the midpoint of a line segment joining the two

points A(x1, y1) and B(x2, y2) are found by averaging the coordinates of the endpoints.

• The midpoint formula is:

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Example 2: F is the midpoint between points C(3, ‒ 4) and D( ‒ 13, ‒ 11). Find its coordinates.

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Solution:

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1 - 4 The Slope of a Line ميقتسملا طخلا ليم

• The slope of a line is a measurement of its steepness and direction.
• Slope of a line m is calculated from the following formula which is

called the slope formula:

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• Depending on the direction of the line, its slope could be positive,

negative, zero or undefined and as shown below.

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Example 3: Find the slope of the line that passes through points P(‒ 4, 8) and R(9, ‒ 7).

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Solution:

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Example 4: Find the slope of the lines a, b, c and d shown in the figure below.

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Solution:

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1 - 5 Parallel Lines ةيزاوتملا طوطخلا

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1 - 6 Perpendicular Lines ةدماعتملا طوطخلا

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Example 5: Lines m and n are parallel. If the slope of line m is ‒ 0.48, what is the slope of line n?

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Solution:

Since the two lines are parallel, then they have the same slope. So, the slope of line n = ‒ 0.48 Example 6: Line c is perpendicular to line d and the slope of line c is 0.5. Find the slope of line d.

Solution:

Since lines c and d are perpendicular, then their slopes are opposite reciprocals of one another. Therefore, the slope of line d =

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1 - 7 x-Intercept and y-Intercept يداصلا عطقملا و ينيسلا عطقملا

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• For a non-horizontal line, x-intercept is the x-coordinate of the point

where the line intersects x-axis.

• In the same way, for a non-vertical line, y-intercept is the y-

coordinate of the point where it intersects y-axis.

a

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1 - 8 Equations of Lines

ميقتسملا طخلا ةلداعم

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• The equation of a line is a mathematical sentence that describes the

relationship between the x-coordinate and the y-coordinate of all its points.

• The equation of a line is of the first degree and is therefore called a linear

equation.

• Straight line equation may be written in any of the following three

forms: where m is the slope, and b is the y-intercept.

where m is the slope.

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Example 7: Draw the graph of the line whose equation is 2x ‒ 3y = 6 using two randomly selected points.

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Solution:

Let x = 1 , then: Thus, (1, ‒1.33) is the first point. Let y = 1 , then: Thus, (4.5, 1) is the second point.

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The graph of 2x ‒ 3y = 6 is as shown in the figure below.

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Example 8: Draw the graph of the line whose equation is 2x ‒ 3y = 6 using the x-intercept and the y-intercept.

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Solution:

x-intercept  y = 0 So, (3, 0) is the first point. Therefore, (0, ‒ 2) is the first point. y-intercept  x = 0

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Example 9: Draw the graph of : (a) x = 4 (b) y = ‒ 2

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Solution:

(a) The graph of x = 4 is a vertical line with x-coordinate = 4 for all its points (b) The graph of y = ‒ 2 is a horizontal line with y-coordinate = ‒ 2 for all its points

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Example 10: Find the equation of the line that passes though the points (4, –5) and (–11, 3). Write the equation in point-slope form, standard form and slope-intercept form.

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Solution:

Find the slope first: Write the equation in point-slope form:

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To write the equation in the standard form, multiply both sides of the equation by 15: Rearrange the equation 8x + 15y = ‒ 43 to write it in the slope-intercept form:  Divide both sides by 15: 

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Example 11: Determine whether the lines 6x + 4y = ‒ 9 and 8x ‒ 12y = ‒ 7 are parallel or perpendicular or neither.

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Solution:

Find the slopes of the two lines and compare them:  m1 =  m2 = So, the two lines are perpendicular since their slopes are opposite reciprocals

• f one another.

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Example 12: Which of the points A(2, 1.6) and B(1, –2.2) lie on the graph of the line 3x + 5y = 14?

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Solution:

If a point lies on the graph of a line then it satisfies its equation.  Point A(2, 1.6) lies on the graph of 3x + 5y = 14 Check point A(2, 1.6): Check point B(1, –2.2) :

 Point B(1, –2.2) doesn’t lie on the graph of 3x + 5y = 14

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1 - 9 Equations of Circles

ةرئادلا ةلداعم

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• The radius r of a circle with a centre at the

point (h, k) can be found using the distance formula between the centre and any point

• n the circle (x, y) and as follows:

(x – h)2 + (y – k)2 = r2

• The standard form of the equation of

a circle of radius r with centre at the point (h, k) is:

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Example 13: The point (3, 4) lies on a circle whose centre is at (‒ 1, 2), as shown in the Figure below. Write the standard form of the equation of this circle.

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Solution:

The radius r of the circle is the distance between (‒ 1, 2) and (3, 4): Using (h, k) = (‒ 1, 2) and r , The equation of the circle is: Standard Form

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1 - 10 Symmetry of Equations تلبداعملا رظانت

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• A graph has symmetry with respect to the y-axis if whenever (x, y) is on the

graph, so is the point (‒ x, y).

• A graph has symmetry with respect to the origin if whenever (x, y) is on the

graph, so is the point (‒ x, ‒ y).

• A graph has symmetry with respect to the x-axis if whenever (x, y) is on the

graph, so is the point (x, ‒ y).

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Example 14: Test y = x 2 + 2 for symmetry with respect to the x-axis, the y-axis, and the origin. .

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Solution:

x-Axis We replace y with ‒ y: The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the x-axis. Multiplying both sides by ‒ 1: y-Axis We replace x with ‒ x: Simplifying gives: The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the y-axis.

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Origin We replace x with ‒ x and y with ‒ y: Simplifying gives: The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin.

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Example 15: Test x 2 + y 4 = 5 for symmetry with respect to the x-axis, the y-axis, and the origin. .

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Solution:

x-Axis We replace y with ‒ y: The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the x-axis. y-Axis We replace x with ‒ x: The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the y-axis.

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Origin We replace x with ‒ x and y with ‒ y: The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the origin.

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