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Analytic Geometry - Equations Involving Lines and Coordinates

1. Point–Slope Form

2. Two-Point Form

3. Slope Intercept Form (Determinant)

4. Slope-Intercept Equation

where $m$ = slope and $b$ = y-intercept.

5. Intercept Form

6. General Equation of a Straight Line

7. Distance from a Point $(x_1, y_1)$ to a Line

8. Distance Between Parallel Lines

For lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$:

9. Angle Between Two Lines Having Slopes $m_1$ and $m_2$

10. Bisector of Angles Between Two Lines

Bisector condition: $d_1 = d_2$.

11. Division of a Line Segment

If $K = \frac{P_1P}{PP_2}$, then:

12. Centroid of a Triangle

If $K = \frac{P_1P}{PP_2}$, then:

For $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$:

13. Slopes of Parallel Lines

14. Slopes of Perpendicular Lines

Thus: $m_2 = -\frac{1}{m_1}$ (m₂ is the negative reciprocal of m₁)

15. Area of Polygons (Coordinate Method)

Related problems will be discussed in the problem sets.

Area (Determinant Method)

Problem Set - Lines and Coordinates

Problem 1:

Write the point-slope form and the slope intercept form of the equation of the line with a slope of 4 that passes through the point (-1,3).

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Problem 2:

A logistic curve is the graph of an equation of the form where k, b, and c are positive constants. Such curves are useful for describing a population y that grows rapidly initially, but whose growth rate decreases after x reaches a certain value. In a famous study of the growth of protozoa by Gause, a population of Paramecium caudata was found to be described by the logitc equation with c=1.1244, k=105, and x time in days. Find b if the intial position was 3 protozoa.
Hint: $y=\frac{k}{1+be^{-cx}}$

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Problem 3:

Find the slope-intercept form of the line that passes through (5,-7) that is perpendicular to the line 6x+3y=4. At what point do these lines intersect?

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Problem 4:

Given A(-3,1) and B(5,4), find the general form of the perpendicular bisector L of the line segment AB.

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Problem 5:

With wireless internet gaining popularity, the number of public wireless internet access points (in thousands) is projected to grow from 2003 to 2008 according to the equation -66x+2y=84, where is is the number of years after 2003. Find the slope and y-intercept of the line equation.

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Problem 6:

The quantity of a product that consumers purchase depends on its price, with higher prices leading to fewer sales. The table below shows the price of a video and the quantity of that video sold on a weekly basis in a store.

Assuming that as price increases, the number sold weekly decreases steadily, use the linear function $y = mx + b$ to find the values of $m$ and $b$, thereby modeling video demand.

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Problem 7:

The equations $5x + 2y = 48$ and $3x + 2y = 32$ represent the money collected from school concert ticket sales during the two class periods. If $x$ represents the cost for each adult ticket and $y$ represents the cost for each student ticket, what is the cost for each adult ticket?

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Problem 8:

Find the area of a triangle whose vertices are A(-3,1), B(5,3), and (2,-8).

Write the coordinates in two columns, x and y. Follow a clockwise or counterclockwise direction when writing the coordinates. In the last row, copy the coordinates of the first row. Then, multiply from left to right and subtract the result when multiplying from right to left. Add all of the values and divide the result by 2. We use an absolute value sign as area cannot be negative.

Problem 9:

What is the angle between the lines y-4x-5=0 and y+2x-1=0?

Note that there are two possible answers because there is an acute and obtuse angle between the two lines.

Problem 10:

Find the distance between the lines 2x+3y=1 and 2x+3y=5.

We may also use the concept of moments in statics of rigid bodies. Treat one line as a unit force and take moments about the second line, vertically or horizontally, depending on preference. The vertical or horizontal distance can be obtained by y₂-y₁ or x₂-x₁. This moment shall equal the unit force multiplied by the perpendicular distance (which is the distance between the two lines).