. Plane and solid analytic geometry; an elementary textbook. the angle between the twTo axes. Sometimes it is easy, as in these cases, to translate thealgebraic equation into the law which governs the move-ment of the point, and hence determine the exact formand position of the locus. But this is often difficult, andwe must have other means of determining the curve. Wecan always determine as many points as Ave please on thelocus by giving to one of the coordinates a series of valuesand determining the corresponding values of the other. 26 ANALYTIC GEOMETRY [Ch. Ill, § 17 Place these points in
. Plane and solid analytic geometry; an elementary textbook. the angle between the twTo axes. Sometimes it is easy, as in these cases, to translate thealgebraic equation into the law which governs the move-ment of the point, and hence determine the exact formand position of the locus. But this is often difficult, andwe must have other means of determining the curve. Wecan always determine as many points as Ave please on thelocus by giving to one of the coordinates a series of valuesand determining the corresponding values of the other. 26 ANALYTIC GEOMETRY [Ch. Ill, § 17 Place these points in their proper positions in the plane,and when a sufficient number has been obtained, a smoothcurve passed through them will show approximately theform of the curve. The points can be determined as nearto each other as we please, and the approximation can becarried to any required degree of accuracy. This is calledplotting the curve. We shall plot the locus of the equation 2 x + y = 10. Give consecutive values to x, and find the correspondingvalues of y. If. x = 0, y = 10. X = 1, y= 8, x=% y = 6, x = 3, y= 4, *=4, y= 2, x = 5, y= o, x = - 1, y=i2, x = - 2, ^ = 14, x = - 3, 2/ = 16, etc etc. Plotting the points (0, 10), (1, 8), (2, 6), andpassing a curve through the points, we see that they allappear to lie on a straight line. This method, however,does not assure us that the locus is a straight line. Itonly shows that, so far as our construction is accurate, itappears to be a straight line. AVe shall show later that every equation of the firstdegree represents a straight line. Ch. Ill, § 18] LOCI 27 Again, let us plot the locus of the equationx*-f= 25. 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. Plo
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