. Plane and solid analytic geometry; an elementary textbook. Ch. IV, § 34] THE STRAIGHT LINE 49 Let MS be this line. Let the inclination of MN be yvand of MS be y2. Then from [11, a J, Solving for ?2, Ave have j _ lr + tan (j> . 2 1-^tanc/)The equation of MS will therefore be If MS is parallel to 3/iV, tan (f> = 0, and the equationbecomes If MS is perpendicular to J/X. tan cf> = oo, and theequation becomes These formulas might be used to write the equations ofparallels and perpendiculars in place of the methods givenin the previous section. PROBLEMS 1. Find the equation of the line th
. Plane and solid analytic geometry; an elementary textbook. Ch. IV, § 34] THE STRAIGHT LINE 49 Let MS be this line. Let the inclination of MN be yvand of MS be y2. Then from [11, a J, Solving for ?2, Ave have j _ lr + tan (j> . 2 1-^tanc/)The equation of MS will therefore be If MS is parallel to 3/iV, tan (f> = 0, and the equationbecomes If MS is perpendicular to J/X. tan cf> = oo, and theequation becomes These formulas might be used to write the equations ofparallels and perpendiculars in place of the methods givenin the previous section. PROBLEMS 1. Find the equation of the line through the origin whichmakes an angle of 60° with the line x — 3 y = 10. 2. Find the equation of the line through (1, 4) which makesan angle of 135° with the line joining (1, 4) with the intersec-tion of 5 x — 2 y = 17 and 3 x -f 4 y = 5. 34. Normal form of the equation of a straight line. — If we have given the length of the perpendicular or nor-mal from the origin on a line, together with the angle 50 ANALYTIC GEOMETRY [Ch. IV, § 34. which this normal makes with the positive direction of theX-axis, the line is completely determined. The perpen-dicular distance is represented by p, and the angle by a. Through 0 draw a linemaking an angle a withOX. If any distance OHis laid off on this lineeither in the positive di-rection (along the termi-nal line of the angle), orin the negative direction,and through H a lineAB, perpendicular to OH,is drawn, that line is com-pletely determined. Itis convenient to restrict a to positive values from 0° to360°. In case we wish to speak of a complete set of parallellines without changing «, it will be necessary to allow p tobe either positive or negative, but every line in the planecan be determined by positive values of both a and p, andthis will always be understood unless otherwise have seen that the equation of the line AB in terms of its intercepts is - -f- j- = 1. But for all positions of the line and Fig. 34. p - = co
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