By A. C. Burdette (Auth.)
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Extra resources for Analytic Geometry
If B is also zero, we define the normal form to be x = 0. The normal form of the equation of a line is particularly useful when the distance from the origin is involved. Example 2-9. Find the equation of the line with normal angle 300° and nor mal distance one. We have, from (2-8), x cos 300° + y sin 300° - 1 = 0 , or -x-y—y-ì=0. 2 2 This is the normal form of the equation of the given line. If, for example, we clear this equation of fractions by multiplying both members by 2, we obtain x-J3y-2 = 09 a perfectly proper equation of the given line but not the normal form.
Thus, by Theorem 3-3, this curve is not symmetric to the origin. t As indicated earlier in connection with the straight line, when there is no chance for misinterpretation we shall use "curve" and "equation" interchangeably. " Also, we shall concern ourselves only with symmetry with respect to the origin and coordinate axes. " Thus we say " a curve is symmetric to the y axis " rather than " a curve is symmetric with respect to the y axis. " 3^. EXTENT 53 Example 3-5. Test x2y - 3x + y = 0 for symmetry.
Then the relationship between the coeffi cients of the general equation and (2-8) may be expressed in the form kA = cos ω, kB = sin ω, kC = -p. From the first two of these we obtain k2A2 + k2B2 = cos2 ω + sin2 ω = 1. Hence 34 2. THE STRAIGHT LINE If C φ 0, we choose the sign so that kC is negative, as we see from the third relation above. In case C = 0, the condition 0 < ω < 180° leads us to choose the sign so kB is positive. If B is also zero, we define the normal form to be x = 0. The normal form of the equation of a line is particularly useful when the distance from the origin is involved.
Analytic Geometry by A. C. Burdette (Auth.)