Back to Analytic Geometry
Analytic Geometry:
AG 005 Equations and Locus:
By William E. Steinman:
There are two things we want analytic geometry to do for us.
First, if we are given an equation, we want to find the corresponding locus.
Or, if we are given a locus defined by some geometric conditions, we want to solve for the corresponding equation.
We need not get hung up on these terms. We know what an equations is.
It is a statement asserting the equality of two expressions.
Well, what is a locus?
We are not talking about an insect here. That is a locust of another kind.
A locus is just a graph, in this case a graph is a curve or straight line containing all of the points whose coordinates satisfy the equation.
We mean just the points which satisfy the equation and no other points.
For now, we can limit this discussion to graphs in just two dimensions, that is graphs represent equations in x and y.
We will need to use some other terms that my not be familiar.
Intercept is one.
An intercept is a point on our coordinate axis where the curve intersects one of the axes. For any given curve, there may be multiple intercepts.
Intercepts are quite easy to find.
In any given equation if we let y = 0 the solution(s) will be the x intercept(s).
Conversely, if we let x = 0 the solution(s) will be he y intercept(s).
For example, in the equation y2 + 4x = 64:
If we let y = 0 we will find that x will be 16.
If we let x = 0 we will find that y will have two solutions at ± 8.
What will this curve look like?
Without assigning numbers or attempting super accuracy, it will look something like this.
Another concept we will need is symmetry:
We know what that means so the formal definition should not trouble us, should it?
Here it is:
Two points are symmetrical with respect to a line if that line is the perpendicular bisector of the line connecting the two points.
Two points are symmetrical about a point if that point is the midpoint of the line connecting the two given points.
Oh dear!
Perhaps it is just as well we drew the above graph.
First, two point are symmetrical with respect to a line if that line is the
perpendicular bisector of the line connecting the two points.
Take a gander at the graph.
We have two points at ±8 on the y axis.
Since they are equidistant from the x axis, that makes the x axis the perpendicular bisector of those two points.
So, by definition, the points are symmetrical.
Well, heck. We know that. It’s obvious.
Also two points are symmetrical about a point if that point is the midpoint of the line connecting the two points.
That means in the above graph, the y intercepts are symmetrical with respect to the origin.
Okay!
What can we surmise from all of that?
If an equation remains unchanged when x is replaced by –x, the graph is symmetrical with respect to the y axis.
Okay, just change our equation from y2 + 4x = 64 to x2 + 4y = 64.
Then the x intercepts will be ± 8 and the y intercept will be 16
That will rotate the graph 900 in the positive direction (clockwise).
When we do that, we can see that the above statement is true.
Also, for every value of y in this equation there are two equal values of x.
By the same token, looking at our original graph we can see that thee following statements are also true.
If an equation remains unchanged when y is replaced by –y the graph is symmetrical with respect to the x axis.
And, for every value of x in this equation there will be two equal values of y with opposite signs.
From all of that, it should follow, if an equation remains unchanged when x is replace by –x and y is replaced by –y the graph will be symmetrical with respect to the origin. Think about it!
Makes sense!
Enough already!