Equations as algebraic objects
Just like numbers, equations can be added or subtracted. Since equations can represent graphs, algebraic operations with equations have interesting consequences for their graphs.
Two lines added or subtracted give another line.
Here we will let L 1 = A 1 x + B 1 y + C 1 so that the equation of line L 1 is L 1 = 0. We use the general form of the straight line for reasons that will become apparent. The sum of two straight lines L 1 = 0 and L 2 = 0 is a third straight line L 1 + L 2 . This so far is not so impressive, but with a new way of writing straight lines it becomes very powerful.
The normal form of a straight line
First we remember that the formula for the distance from a point (x
1
, y
1
) to a line is given by the formula on the right. Using this as a model we can write the straight line in a new and powerful way. Divide the general form of a straight line by √(A
2
+ B
2
) obtaining L
1
= a x + by + c where a, b, and c are given to the right. L
1
= L
1
(x,y) is a function of x and y. The amazing thing is that
L
1
(x
1
, y
1
) is the directed distance from the line to point (x
1
, y
1
)! By using this form the algebraic expression of a straight line both represents the graph of a straight line and a distance from the line. If
(x
1
, y
1
)
is on the line then and we have the equation of the line. If
(x
1
, y
1
)
is not on the line then | L
1
(x
1
, y
1
)
| is the distance to the line. Note: if we leave off the absolute values, then L
1
(x
1
, y
1
)
is the directed distance, the sign indicating on which side of the line the point resides. Directed distances give more information and are thus more useful.
The angle bisectors of two lines
If L 1 and L 2 are two lines in normal form, then L 1 + L 2 and L 1 – L 2 are the two angle bisectors. Proof: let P be on the bisector. Then | L 1 (x 1 , y 1 ) | = | L 2 (x 1 , y 1 ) | giving either L 1 (x 1 , y 1 ) – L 2 (x 1 , y 1 ) = 0 or L 1 (x 1 , y 1 ) + L 2 (x 1 , y 1 ) = 0 as the equations of the bisectors.
Adding or subtracting circles
If a circle C 1 is written in normal form C 1 = x2 + y 2 + Ax + By + C. If two circles are written in this form, then C 1 – C 2 is a straight line. This straight is called the radical axis of the two circles. The power of any point on the radical axis with respect to each circle is the same.
If two circles intersect, then the radical axis is the l
ine joining the two point of i
ntersection
. The radical axis is always perpendicular to the line joining the two centers. If the circles do not intersect, then the radical axis is closer to the larger circle.
Geometric formulas are more true than you think
As we have seen above, functions that define graphs can still have meaning if evaluated at points not on the graph. We showed that a function describing a straight line in the normal form can give the distance from the point to the line when the point is not on the line. Here are two more examples of this using circles
Consider the circle described by (x-h) 2 + (y-k) 2 - r 2 . If the coordinates (x,y) are on the circle of radius r, centered at (h,k), this function is zero. If P is somewhere else, then this expression is the power of the point with respect to the circle. If furthermore the point is outside the circle, then the power of the point is the square of the tangent line to the circle.
The tangent to x 2 + y 2 = r 2 at (x 1 , y 1 ) is given by the simple equation x 1 x + y 1 y = r 2 . If the point is not on the circle, the line is then called the polar of the point P.
x
x
x
x
x
x
x
x