By A. C. Burdette (Auth.)
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Additional resources for Analytic Geometry
Then, from the first equation, sin ωί = — 2 cos cou which substituted in the second gives 4 cos2 ωγ + cos 2 ωί = I, or cos ft)! = ± ν ϊ · Noting that ωι is a second-quadrant angle, we have cos ωγ = 1 — and d'il /"lì sin coi =■ Then Ιγ has the equation x 2v - - p + -7=-2=0. x/5 ^5 For / 2 , we have cos ft)2 = c o s ( œ 1 + 180°) = - s i n ft^ = — p , sin ω 2 = sin (ω! + 180°) = — sin ft^ = 2 —, 36 2. THE STRAIGHT LINE and its equation may be written x 2y -2=0. Example 2-11. Find the distance from the origin to the line 2x + 3>> + 10 = 0.
Distance from a Line to a Point Another use of the normal equation is to find the directed distance from a line to a point. This does not require any definition of directions because a positive direction has already been assigned on the normal to a line. For example, we see in Figure 2-11 that dl is negative because it is measured opposite the positive direction on the normal. For similar reasons d2 is positive. In general, 38 2. THE STRAIGHT LINE , v, "V. Pi \ y \ l \ 0 Pi X P(x» yù Figure 2-11 Figure 2-12 if P and the origin are on opposite sides of the line, the directed distance will be positive ; if they are on the same side, it will be negative.
Lfy = 0,x= ± 2 . t In order to emphasize the role of excluded values, that portion of the plane from which the curve is excluded is shaded. 3-6. GRAPHING EQUATIONS 57 (-5,3) Figure 3-4 (b) Symmetry: Theorems 3-1-3-3 all indicate symmetry. f (c) Extent: From Example 3-8, |JC| < 2 is excluded. (d) Horizontal and vertical asymptotes: Section 3-5 gives no horizontal or vertical asymptotes. (e) Additional points: The results of (a)-(d) make it possible to sketch the required graph from very few additional points.