. Plane and solid analytic geometry . the single equation l{x + y-2)^m{x-y)=0, where I and m have arbitrary values, not both zero. 170 ANALYTIC GEOMETRY In general, if u = 0 and v = 0 are any two curves, all thecurves represented by the equation lu -f- w-^J = 0, where I and m have arbitrary values, not both zero, form whatis called a pencil of curves. Applications. Example 1. Let u = X- -\- if -\- ax -\- hy -\- c = Ojy = aj^ + ?/2 -f ax + hy + c = 0, be the equations of any two circles which cut each the equation u-v={a-a)x+{h-h)y+{c-c)=0 represents a curve which passes through the
. Plane and solid analytic geometry . the single equation l{x + y-2)^m{x-y)=0, where I and m have arbitrary values, not both zero. 170 ANALYTIC GEOMETRY In general, if u = 0 and v = 0 are any two curves, all thecurves represented by the equation lu -f- w-^J = 0, where I and m have arbitrary values, not both zero, form whatis called a pencil of curves. Applications. Example 1. Let u = X- -\- if -\- ax -\- hy -\- c = Ojy = aj^ + ?/2 -f ax + hy + c = 0, be the equations of any two circles which cut each the equation u-v={a-a)x+{h-h)y+{c-c)=0 represents a curve which passes through the two points of intersection of the circles. But this equation, being linear, represents a straight line, and is, therefore, the equation of the common chord of the circles. The foregoing proof is open to the criticism that conceivably we might have a-a = 0, b-b = 0, and then the equation u —v = 0 would not represent a straightline. But in that case the circles would be concentric, and we have demanded that they cut each other. Wi=0. «i=0 Example 2. We can now provethe following theorem : Given threecircles, each pair of which their three common chordspass through a point, or are parallel. Let two of the three given circlesbe those of Example 1, and let theequation of the third circle be w = x^-\-y^ + ax + by + c — 0. CERTAIN GENERAL METHODS 171 Then the equations of the three common chords can be writtenin the form: u — v = 0, V— ^o = 0, w— u = 0. Let We observe that the equation, (8) Ui-\-Vi-\- ivi = 0 or — Wi = %i + Vi, holds identically for all values of x and y. Consequently, theline Wi = 0 is the same line as Ui + Vi = 0, and therefore it passes through the point of intersection ofUi = 0 and Vi = 0, or, if these lines are parallel, is parallel tothem. Hence the theorem is proved. The above proof is a striking example of a powerful methodof Modern Geometry known as the Method of Abridged Nota-tion.* By means of this method many theorems, the proof
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