Plane and solid analytic geometry; an elementary textbook . x*-y* Solving the equation for y, we have y = ± Va?2 — 25,from which it appears that y is imaginary, so long as— 5 < x < + 5. There will therefore be no points on thelocus for which x is numerically less than 5. If x — 5, y — 01 x = — 5, y — 0; x = 6, y = ±VIlj x = -6, ?/ = ±Vll; # = 7, y == ± V24 etc. Plotting the points (5, 0), ((3, + Vll), (6, - Vll), 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 e
Plane and solid analytic geometry; an elementary textbook . x*-y* Solving the equation for y, we have y = ± Va?2 — 25,from which it appears that y is imaginary, so long as— 5 < x < + 5. There will therefore be no points on thelocus for which x is numerically less than 5. If x — 5, y — 01 x = — 5, y — 0; x = 6, y = ±VIlj x = -6, ?/ = ±Vll; # = 7, y == ± V24 etc. Plotting the points (5, 0), ((3, + Vll), (6, - Vll), 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 ?/, we find two values of x, equalnumerically but with opposite signs, the curve is evi-dently symmetrical with respect to the T^-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 ivhen its equationdoes not contain odd powers of x. It is symmetrical with respect to the origin if its equationcontains no term o
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