Back to Analytic Geometry
Analytic Geometry:
AG 006 The Straight line:
By William E. Steinman:
Let’s go way back in geometry to our old friend the straight line.
A straight line is the graph or locus of an equation in the first degree of two variables.
A straight line is completely determined if its direction is given along with a point through which the line must pass.
We can have the equation in the point slope form.
For example y – y1 = m(x – x1), where m is the slope.
This is the equation of a straight line though point (x1, y1)
Just for review:
The slope is the rate at which an ordinate of a point of a line on a coordinate plane changes with respect to a change in the abscissa.
The abscissa is the coordinate representing the position of a point along a line perpendicular to the y axis in a plane Cartesian coordinate system. In he above equation that is x1.
The ordinate is the point in the plane Cartesian coordinate system representing the distance from a specified point to the x-axis, measured parallel to the y axis. In the above equation that is y1.
So the slope becomes the rate of change which is (y – y1)/(x – x1)
Okay?
We can have the slope intercept form.
y = mx + b where m is the slope and b is the y intercept.
We can have the two point form.
That is a straight line which passes through two defined points.
(y – y1)/(x – x1) = (y1 – y2)/(x1 – x2)
We have the intercept form where the x and y intercepts are (a,0) and (b,0) respectively.
The resultant equation is x/a + y/b =1
In the general form then, every equation of the first degree in x and y can be reduced to the form:
ax + by + c = 0. a, b and c are, of course, arbitrary constants.
For this form the slope (m) = -a/b.
The y intercept is –c/b.
Now, in looking at the above definitions and the next statement, I have realized I did not directly address the idea of slope in my previous study notes. This is one of those concepts we sometimes cover in algebra and sometimes we do it early I geometry. I did neither, so let us quickly correct that by stating most of the principles involved in the slope of a line.
1: If a line passes through points (x1, y1) and (x2, y2) then slope = (y2 – y1)/(x2 – x1)
That is the change in y divided by the change in x.
2. The slope of a line is equal to the tangent of its inclination.
3. If a line slants upward from left to right its inclinations is an acute angle and its slope is positive.
4. If a line slants downward from left to right its inclination is an obtuse angle and its slope is negative.
5. If a line is parallel to the x-axis its inclination is 0 and its slope is 0.
6. If a line is perpendicular to the x axis its inclination is 900 and it has no slope.
7. Parallel lines have the same slope.
8. Perpendicular lines have slopes that are the negative reciprocals of each other.
Okay, now we can go on.
A straight line is completely determined if the length of the perpendicular from the origin to the line is known and the angle which the perpendicular makes with the x axis is known.
Let’s have a look.
Consider the line (a,b) which passes through point c.
The perpendicular P is the line to a,b from the origin to c.
The angle of the perpendicular with the x axis is given as z. Now from trigonometry we know that:
x1 = P cos z
y1 = P sin z
The slope of (ab) = m = -1/tan z =-cot z = -cos z/sin z.
It’s just another walk in the park.
Now, if (x, y) is any other point on our line (a,b), by the point slope formula we have (y – y1) = - cot z(x – x1)
Or y – P sin z = -(cos z/sin z)(x – P cos z)
Can we reduce this to the standard form?
If it’s math and we don’t make any errors it should be a cake walk.
First, lets multiply the right side out.
y – P sin z = (-x cos z/sin z) + (P cos2 z/sin z)
Okay, lets move it all to the left side.
(x cos z/sin z) + y – P sin z – (P cos2 z/sin z = 0
Can we multiply both sides by sin z? Sure!
x cos z + y sin z –P sin2 z – P cos2 z = 0
Lets factor out –P
X cos z + y sin z –P(sin2 z + Cos2 z) = 0
But that sin2 + cos2 is a trigonometric identity equal to 1.
Shazam!
I’ll let you write the last line.