Elements of analytical geometry and the differential and integral calculus . THE CIRCLE. 37 Application.—The polar equation of the circle in its mostgeneral form is r^-\-2{{)r+a^+b^=E^. (1) If we make 5=0, it puts the polar point somewhere on theaxis of X, and reduces the equation to r^+ga cos. +a2=i23. (2) Now if we make v=0, thenwill cos. v=l, and the linesrepresented by ±r would refer tothe points X, X, in the circle. This hypothesis reduces the lastequation to r^+2ar=(E^—a=) (3)and this equation is the same inform as the common quadratic inalgebra, or in the same form as x^±
Elements of analytical geometry and the differential and integral calculus . THE CIRCLE. 37 Application.—The polar equation of the circle in its mostgeneral form is r^-\-2{{)r+a^+b^=E^. (1) If we make 5=0, it puts the polar point somewhere on theaxis of X, and reduces the equation to r^+ga cos. +a2=i23. (2) Now if we make v=0, thenwill cos. v=l, and the linesrepresented by ±r would refer tothe points X, X, in the circle. This hypothesis reduces the lastequation to r^+2ar=(E^—a=) (3)and this equation is the same inform as the common quadratic inalgebra, or in the same form as x^±px= x—r, ^a=d=:p, and jf^^—a^= These results show us that if we describe a circle with the radius Jq-^-lp^, and place F on the axis of X at a distancefrom the center equal to ^p, then PX represents one value ofX, and FX the other. That is, ^=-lP+j9+lP=J^ Or x^=^\p—Jx-\-ip^^=:FX\ and this is the common solution. When p is negative, the polar point is laid off to the left fromthe center at F\ The operation refers to the right angled triangle AFM, JF=±p, FM=^q, and AM=Jq-{-\pKLet the form of the quadratic be X^zhpX: ^;3 ANALYTICAL GEOMETRY. Then comparing this with the polar equation of the circle, wehave
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