Download Analytic Geometry by A. C. Burdette (Auth.) PDF

By A. C. Burdette (Auth.)

ISBN-10: 0121422569

ISBN-13: 9780121422561

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We could solve this problem by the method of Example 2-8. However, it is most easily solved by reducing the equation to normal form and reading the normal distance p directly from this equation. From (2-9), making the appropriate choice of sign, k= 1 -- 2 1 2 Jl + 3 Vl3 Then the normal form of the given equation is 2 3 10 Vl3 Vl3 ^13 from which we immediately obtain the normal distance p = 10/^/13. As a by-product, however, useless in this case, we have (Figure 2-10) sin ω = — · x/l3' cos ω = — y13' From these equations, it is possible to determine the normal angle ω.

Test x2y - 3x + y = 0 for symmetry. The student can easily verify that the simultaneous substitution of — x for x and — y for y leads to an equivalent equation, but either of these changes alone produces a nonequivalent equation. Hence this curve is symmetric to the origin, but not to either axis. The application of the concept of symmetry to the problem of graphing an equation is obvious. 3-4. Extent A valuable aid in drawing the graph of an equation is knowledge of the extent of the curve, that is, what portion or portions of the plane are occupied by the curve.

Hence we must have 4-2 = k(-3- 1), or k = —\. Thus the required equation is y-2=-i(x-l\ or x + 2y - 5 = 0. This, of course, is precisely what we would have obtained from the twopoint equation. Example 2-16. Find the equation of all lines parallel to 4x — 3y + 15 = 0 and 3 units distant from (3, — 1). First we write the equation of the family of lines parallel to the given line. Obviously this may be written 4x - 3y + k = 0. 2-12. FAMILIES OF LINES THROUGH THE INTERSECTION OF TWO LINES 45 Now we wish to pick the members of this family which pass at a distance of 3 units from (3, - 1 ) .

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