By A. C. Burdette (Auth.)

ISBN-10: 0121422569

ISBN-13: 9780121422561

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Sample text

THE NORMAL EQUATION OF THE STRAIGHT LINE 31 5. Find the equations of the sides of the triangle whose vertices are (— 1, 8), (4,-2), ( - 5 , - 3 ) . 6. Find the equations of the medians of the triangle given in Exercise 5. 7. Find the equations of the altitudes of the triangle given in Exercise 5. 8. Find the equation of the line that has —3 and 5 for its x and y intercepts, respectively. 9. Find the equation of the line that has the y intercept 4 and is parallel to the line 2x — 3y = 7. 10. Find the equation of the line that has the x intercept — 3 and is perpen­ dicular to the line 3x + 5y — 4 = 0.

A) We have, from the slope-intercept equation, y = 2x + k. (b) The x intercept 3 implies that members of the family all pass through (3, 0). Hence, from the point-slope equation, y = k(x — 3). 44 2. THE STRAIGHT LINE (c) The slope of the given line is f. Hence all lines perpendicular to it have the slope —-§-. Thus, from the slope-intercept equation y = —\x + k. Many problems involving the determination of the equation of a straight line may be solved by using appropriate families of lines. The following two examples will illustrate the central concept in many of these.

We have ωχ = α, ω 2 = a + 180°. Instead of finding ω, it will serve our purpose equally well to find sin ω and cos ω. We can obtain two equations in the unknowns sin ωχ and cos ωγ by writing sinw, cos ωί and = -2, sin2 col + cos2 coj = 1. 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.