By Luther Pfahler Eisenhart
This textbook explores the configurations of issues, strains, and planes in area outlined geometrically, interprets them into algebraic shape utilizing the coordinates of a consultant element of the locus, and derives the equations of the conic sections. The Dover version is an unabridged republication of the paintings initially released through Ginn and corporate in 1939.
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Extra info for Coordinate Geometry
Consequently we have the theorem An equation of a straight line is of the first degree in x and y. We say "an equation/' and not "the equation/' because, if we have an equation of a line, so also is any constant multiple of this equation an equation of the line, since any solution of a solution of the other. Having shown that in an equation of any line the coefficients of x and y are not both simultaneously equal to zero, we remark that if, as stated in 1, in deriving theorems in this chapter we exclude from consideration the degenerate case of equations of the first degree for which the coefficients of both x and y are zero, we are not thereby restricting the consideraeither equation tion of all is the lines in the plane.
11. 8) for x and y are substituted in the value of hi/h 2 obtained from the result is the "ratio in which the segment PiP* is divided by the point of intersection of the line ax + by c and the line through PI and 12. an equation ax if + by + c = 0, + = P2 The . reader should apply this method to Ex. 26 7. Sec. 6] Numbers and Direction 6. Direction Numbers and Direction Cosines of a Line Direction Cosines of a Line. Angle between Two Lines The equation (xi, yi) and (x 2 y 2 ), obtained in the condition that the direction numbers of the expresses and are PiP 2 P\P segments proportional when P is any point on the line.
10, X is positive. 1) and this is when inclined as in Fig. 11, time take PI below the line, add variety to the discussion. Comparing the line PiP 2 in Fig. 11 with the line PiP in Fig. 5) _x ^ where d is - dx FlG< =d n a positive number, the directed distance PiP 2 . 3), we see that the left-hand members of these respective equations differ only in sign. 3) apply in this case also, with the understanding that d is negative; it is the directed distance P 2 Pi, that is, PI is on the negative side of the line.