ANALYTICAL GEOMETRY

SJ van Heerden 09/03/2014

HISTORYIntroduced in the 1630s

Aided the development of calculus

RENE DESCARTES (1596-1650) and

PIERRE DE FERMAT (1601-1665),

French mathematicians,

independently developed

the foundations for

analytical geometry

CARTESIAN PLANE

• x-axis (horizontal axis) where the x values are plotted along.

• y-axis (vertical axis) where the y values are plotted along.

• origin, symbolized by 0, marks the value of 0 of both axes

• coordinates are given in the form (x,y) and is used to represent different points on the plane.

Slope of a Line

Slope of a Line

• If line rises from left to right,

Slope of a Line

• If line goes from right to left

Slope of a Line

• If line is parallel to x-axis

Slope of a Line

• If line is parallel to y-axis

Inclination of a Line

O M

θx

y

L

O M

θx

y

L

Angle between Two Lines

Angle between Two Lines

• If θ is angle, measured counter-clockwise, between two lines, then

• where m2 is the slope of the terminal side and m1 is the slope of the initial side

SLOPE OF PARALLEL LINES

• Two non-vertical lines and are parallel if and only if their slopes are equal and the angle of inclination are also equal.

(slope)

(angle ofinclination)

SLOPE OF PERPENDICULAR LINES

• Two non-vertical lines and are perpendicular if and only if their slopes are negative reciprocals of each other.

or

or

•

c. INTERCEPT FORM - the intercept form is given by the equation

d. TWO-POINT FORM - the two point form equation of the line is given by

• The general form of the equation of a line is given by

where A, B and C are non-zero constants and x and y

are variables of degree one.

Sample Problems•

3. Show by means of slope that the points , , and are the vertices of a parallelogram.

4. Show by means of slope that the points , and are vertices of a right triangle.

5. Find the angle from the line through the points and to the line through the points and .

6. The angle between two lines and is . If the gradient of is , find the gradient of .

1. Find the general equation of the line through with slope 4.

2. Find the general equation of the line passes through the point and .

3. Find the general equation of the line having gradient of and y-intercept of .

4. Reduce the equation of to the slope intercept form.

5. Find the general equation of the line having x-intercept of 3 and y-intercept of -2.6. Find the general equation of the line passes through the point and with equal intercepts.7. Determine the general equation of the line passes through the following points:

a. b.

1. Find the equation of the line parallel to through the point .

2. Find the equation of the line through which is perpendicular to the line which passes through the points and .

3. Show that lines and form a rectangle.

EQUATION OF A CIRCLE

• Using distance formula to find the distance between points A and B, we have

(1)• Since the distance between points A and B is the radius of

the circle, we will let d = r. Thus, (2)

• Point A is in the origin, therefore , then

(3)

• From (3), we can say that the equation of circle having its center at the origin is,

B

A

EQUATION OF CIRCLE (origin is not the center)

A (h, k)

B ( x, y)

• The distance between points A and B is still the radius of the circle. So,

(5)• Thus, (5) is the equation of the circle if the center is at (h,

k). • Expanding (5), and letting, it will lead to the general

equation of the circle. Thus,

(6)

Graph the following circle.

REFERENCES

- Felipe, N, M. (2013). Analytic geometry basic concepts [PowerPoint Presentation]. Available at: http://www.slideshare.net/NancyFelipe1/analytic-geometry-basic-concepts.Accessed on: 4 March 2014

- Marasigan, D. (2013). Lecture #2 analytic geometry [PowerPoint Presentation]. Available at: http://www.slideshare.net/denmarmarasigan/lecture-2-analytic-geometry.Accessed on: 4 March 2014

- Marasigan, D. (2013). Lecture #3 analytic geometry [PowerPoint Presentation]. Available at: http://www.slideshare.net/denmarmarasigan/lecture-3-analytic-geometry.Accessed on: 4 March 2014

- Marasigan, D. (2013). Lecture #5 analytic geometry [PowerPoint Presentation]. Available at: http://www.slideshare.net/denmarmarasigan/lecture-5-analytic-geometry. Accessed on: 4 March 2014

- Demirdag, M. (2013). Analytic geometry [PowerPoint Presentation]. Available at: http://www.slideshare.net/mstfdemirdag/analytic-geometry-8693115. Accessed on: 4 March 2014