Understanding Lines and Angles in Analytic Geometry
Length of a Segment from a Point to a Line
The shortest distance from a point P (x1, y1) to a line Ax + By + C = 0 is given by the following expression:
In this expression, (x,y) represent the point from which we want to measure the distance, and A, B, and C are the coefficients of the line.
The conditions to consider are the following:
- If the value of C is not equal to 0, the sign of the radical is the opposite of C.
- If C=0 and B is not equal to 0, the radical will have the same sign as B.
- If C=B=0, the radical and A will have the same sign.
Example
Determine the equation of the line that contains the point (2, 1), knowing that there is a minimal distance between this point and point (4, 3).
Steps
- Procedure
- Determine the distance between both given points.
- Calculate the slope of the line that represents the distance between the points.
- Consider that the calculated slope is from the line perpendicular to the line from which we want to obtain an equation.
Then, based on the perpendicularity theorem, we consider the slope calculated in the previous step as reciprocal and with the opposite sign:
Using the point-slope equation of the line, we obtain the requested equation.
To prove, we obtain the distance from point (4, 3) and the line that resulted from the previous step.
Rationalize the resulting value.
This is the same distance between the points obtained in the first step.
Answer
The equation of the line is:
Angle Between Two Lines
Let A1x + B1y + C1= 0 and A2x + B2y + C2 = 0 be two lines that intercept at one point and form an angle 
obtained by the following expression:
Where m1 is the slope from which the angle is first measured and m2 is the slope of the line where the angle is finally measured. The angle is measured counterclockwise.
Example
Calculate the angle formed between the lines 3x-4y+5=0 and x+6y-3=0. Graph the lines and the angle between them.
Steps
- Procedure
- First, graph the lines for determining the order of the slopes in the angle’s formula. In order to graph them, obtain the intersections with each of their axes.
- Sketch the obtained points
- Find the slopes of the lines by rearranging the equation as slope-ordinate form.
- Substitute the slopes in the formula to determine the angle.
Answer
Equation of a Family of Lines
The equations of the line have been expressed in different forms:
Each of them has two constants with a specific geometric importance, known as parameters. For example, in the first equation, the parameters are m and b, while in the second, the parameters are a and b.
A linear equation with one parameter represents a group of lines that have the same characteristic, for example:
In this example, the equation represents all those lines that cut the y axis at y=5, regardless of its inclination angle or slope.
Example
Write the equation of the family of lines that pass through point (5, 3).
Steps
- Procedure
- Substitute the given point into the equation.
- Multiply the m value by the expression contained within the parenthesis.
Answer
The equation of the family of lines that pass through point (5, 3) is: 
where m is the value of the slope.
Polar Form of the Equation of a Line
The general equation of the straight line 
can be transformed into its polar form by substituting 
and 
into the following equation:
The equation of the line that passes through the origin as the form 
will be expressed as:
Example
Transform the equation 6x – 2y + 8 = 0 into its polar form.
Steps
- Procedure
- Substitute the x and y values in the equation by their relation in polar form.
- Solve for r in the expression, in order to obtain the polar equation of the line.
Answer
