( ii \ri u: in 



LOCUS OF AN EQUATION 



32. The locus of an equation. A pair of numbers , jr is 



rented gecim-tii- -ilU by a point in a plane. If these 



numbers (z, y) are variables, but connected by an equa- 



, th.-u this equation can, in general, be satisfied by an 



m hint.- number of pairs of values of x and y, and each pair 



may be represented by a point. These points will 



<-ver, be scattered indiscriminately over the plane, )>ut 



\\ill nil lie in a definite curve, whose form depends only 



iijM.ii the natuiv f tin- r.|u.iti>u undt-r consideration; and 



\e will contain no points except those whose co- 



uates are pairs of values which when substituted f<r 



m and y, sati>t\ the given equation. This curve is called 



the locos or graph >f the equation; an<l tin- first fuinl.i- 



mental problem of analytic geometry is to find, fur a gi\cn 



equation, its graph or locus. 



33 Illustrative examples : Cartesian coordinates. 



en tke equation x + 5 = 0, to find its locus. This equation is 

 tiffed by the pairs of values ^=-5, fl = 2; ^ = -5, f,= *; 

 - 5, y, = - 2 ; etc n that i*, by every pair of values for which x = 5. 

 Such points as 



/>, = (,,.*) a (-5, 2), 



all lie on the line Jl/.V. parallel to the y^ra. and at the distance 5 on 

 the negative side of it, this line ertesislinf indefinitely in both 



48 



