An Elementary Course in Analytic Geometry
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79. In rectangular coordinates every equation of the form cc -+ 2/'^ + 2 Goc + 2 i^'i/ + C = represents a circle. The equa- tions of the circles already obtained (equations [31] and [32], as well as the answers to examples 1 to 5 and 7) are all of the form x^^-y'^ + 2ax+2Fy-{-C=0; . . . (1) it will now be shown that, for all values of ^, F^ and (7, for which the locus of equation (1) is real, this equation represents a circle.
To prove this it is only necessary to complete the square in the ^-t...erms and in the ?/-terms, by adding G^ -f F^ to each member of equation (1), and then transpose C to the second member. Equation (1) may then be written in the form (x + G-y + (?/ + Ff = a'^ + F^- = (V(/2 + i^2_ (7)2 . , (2) which is (cf. equation [31]) the equation of a circle Avhose center is the point (— (r^, — ^), and whose radius is 138 ANALYTIC GEOMETRY [Ch. VII.
Note 1. This circle is real only ii G^ + F^ - C>0; for, if G^+F-2- C
What to read after An Elementary Course in Analytic Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Tanner, J. H. (John Henry), B. 1861 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
79. In rectangular coordinates every equation of the form cc -+ 2/'^ + 2 Goc + 2 i^'i/ + C = represents a circle. The equa- tions of the circles already obtained (equations [31] and [32], as well as the answers to examples 1 to 5 and 7) are all of the form x^^-y'^ + 2ax+2Fy-{-C=0; . . . (1) it will now be shown that, for all values of ^, F^ and (7, for which the locus of equation (1) is real, this equation represents a circle.
To prove this it is only necessary to complete the square in the ^-t...erms and in the ?/-terms, by adding G^ -f F^ to each member of equation (1), and then transpose C to the second member. Equation (1) may then be written in the form (x + G-y + (?/ + Ff = a'^ + F^- = (V(/2 + i^2_ (7)2 . , (2) which is (cf. equation [31]) the equation of a circle Avhose center is the point (— (r^, — ^), and whose radius is 138 ANALYTIC GEOMETRY [Ch. VII.
Note 1. This circle is real only ii G^ + F^ - C>0; for, if G^+F-2- C
What to read after An Elementary Course in Analytic Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Tanner, J. H. (John Henry), B. 1861 to read onlineMoreLess
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