By Luther Pfahler Eisenhart
A radical, whole, and unified creation, this quantity gives remarkable insights into coordinate geometry. Invariants of conic sections and quadric surfaces obtain complete remedies. Algebraic equations at the first measure in and 3 unknowns are rigorously reviewed. through the ebook, effects are formulated accurately, with truly said theorems. greater than 500 beneficial workouts. 1939 version.
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Additional resources for Coordinate Geometry (Dover Books on Mathematics)
4) In like manner, by 0i if may be written 0i Ci 02 c2 = 0. 2) result subtract the first multiplied 0i 0i 02 02 40 = 0, C2 by 2, Determinants of Sec. 6) are definitions. 4) and in We minants. 5) are and the obtained from the fr's by c's first on replacing the a's by c's respectively. 5) in detail. Suppose that the 0's and &'s have such values that the determinant ai 1 $2 U2 is Then equations not equal to zero. 5) can be solved at once for x\ and y\, the common solution; this is the process with which the reader is familiar, although maybe not in this notation.
10) by (+) and ( ) respectively. 10) with the sign is that of the bisector whose points lie above one line and or e 2 a 2 0. 10) is that of the with the sign bisector whose points lie above, or below, one line and to the left, or right, of the other, as the reader should verify. ; For example, the bisectors of the angles between the lines have the respective equations (2 V5 4)* + V5 T 3)y ( 51 (3 V5 1) = 0. y-l=0 and the point (1, - 1) 1. lines ; of the line through the intersection ( 2, 7) ; of the through the intersection and parallel to the *-axis.
3, show that b\c 2 -f- is a\d\ 4- b\d% I b2 c 2 equal to the product of the determinants 46 ai l . *' The Sec. 10] Set of Lines through a Point The Set 10. 7) holds. Thus two intersecting lines determine a point. This is called the dual of the theorem that two points determine a line. 3) gives the relation between the coordinates x, y of any point of the line determined by the points (xi, ;yi) and (x 2 y 2 ) and the coordinates of these two points. As the dual of this result there should be a relation between an equation of any line through a point and equations of two lines determining the point.