. Plane and solid analytic geometry; an elementary textbook. 5. Solving the equation for y, we have y = ± Vz2 — 25,from which it appears that y is imaginary, so long as— 5 < x < -f- 5. There will therefore be no points on thelocus for which x is numerically less than 5. If x = 5, y = 0 ; x = — 5, y = 0 ; a; = 6, */ = ±VTl; a = — 6, 2/ = ±VlT; x—1, y = ± V24 ; etc. Plotting the points (5, 0), (6, + VII), (6, - VTT), etc.,and passing a smoothcurve through them, we \ y have the curve in It can be seen fromthe equation that eachbranch goes off indefi-nitely, never again turn-ing towa
. Plane and solid analytic geometry; an elementary textbook. 5. Solving the equation for y, we have y = ± Vz2 — 25,from which it appears that y is imaginary, so long as— 5 < x < -f- 5. There will therefore be no points on thelocus for which x is numerically less than 5. If x = 5, y = 0 ; x = — 5, y = 0 ; a; = 6, */ = ±VTl; a = — 6, 2/ = ±VlT; x—1, y = ± V24 ; etc. Plotting the points (5, 0), (6, + VII), (6, - VTT), etc.,and passing a smoothcurve through them, we \ y have the curve in It can be seen fromthe equation that eachbranch goes off indefi-nitely, never again turn-ing toward either axis;for as x increases, y in-creases indefinitely. 18. Symmetry. — A curve is said to be symmetricalwith respect to one of two axes (rectangular or oblique)when that axis bisects every chord parallel to the other. A curve is said to be symmetrical with respect toa point when that point bisects every chord drawnthrough it. It is easily proved that if a curve is symmetrical withrespect to two axes, it is symmetrical with respect to. 28 ANALYTIC GEOMETRY [Ch. Ill, § 18 their point of intersection. Now, if, upon substitutingany value for x in an equation, we find two values of #,equal numerically but with opposite signs, the curve isevidently symmetrical with respect to the X-axis. Or,if, for every value of y, we find two values of x, equalnumerically but with opposite signs, the curve is evi-dently symmetrical with respect to the Y-axis. If boththese occur, the curve must be symmetrical with respectto the origin. It appears that the first of these conditions can besatisfied when y occurs in the equation in even powersonly, and the second when x occurs in even powers curve- is therefore symmetrical with respect to the X-axistvhen its equation does not contain odd powers of y; it issymmetrical with respect to the Y-axis when its equationdoes not contain odd powers of x. It is symmetrical with respect to the origin if its equationcontains no term of
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