Last updated at Feb. 3, 2020 by Teachoo
Transcript
Ex 10.1, 11 The slope of a line is double of the slope of another line. If tangent of the angle between them is 1/3 , find the slopes of the lines. Let m1 & m2 be the slopes of two lines We know that angles between two lines are tan ฮธ = |(๐2 โ ๐1)/(1 + ๐1๐2)| Here tan ฮธ = 1/3 & m2 = 2m1 Putting values tan ฮธ = |(๐2 โ ๐1)/(1 + ๐1๐2)| 1/3 = |(2๐1 โ ๐1)/(1 + ๐1(2๐1))| 1/3 = |๐_1/(1 + 2ใ๐_1ใ^2 )| |๐_1/(1 + 2ใ๐_1ใ^2 )| = 1/3 So, ๐_1/(1 + 2ใ๐_1ใ^2 ) = 1/3 or ๐_1/(1 + 2ใ๐_1ใ^2 ) = ( โ1)/3 Solving ๐_๐/(๐ + ๐ใ๐_๐ใ^๐ ) = ๐/๐ 3m1 = 1 + 2ใ"m1" ใ^2 2ใ"m1" ใ^2 + 1 โ 3m1 = 0 2ใ"m1" ใ^2 โ 3m1 + 1 = 0 2ใ"m1" ใ^2 โ 2m1 โ m1 + 1 = 0 2m1(m1 โ 1) โ 1(m1 โ 1) = 0 (2m1 โ 1) (m1 โ 1) = 0 So, m1 = ๐/๐ , m1 = 1 Solving ๐_๐/(๐ + ๐ใ๐_๐ใ^๐ ) = (โ๐)/๐ 3m1 = โ1 โ 2ใ"m1" ใ^2 2ใ"m1" ใ^2 + 1 + 3m1 = 0 2ใ"m1" ใ^2 + 3m1 + 1 = 0 2ใ"m1" ใ^2 + 2m1 + m1 + 1 = 0 2m1(m1 + 1) + 1(m1 + 1) = 0 (2m1 + 1) (m1 + 1) = 0 So, m1 = (โ๐)/๐ , m1 = โ1 When m1 = ( ๐)/๐ m2 = 2m1 m2 = 2(1/2) = 1 When m1 = 1 m2 = 2m1 m2 = 2(1) = 2 When m1 = ( โ๐)/๐ m2 = 2m1 m2 = 2(( โ 1)/2) = โ1 When m1 = โ1 m2 = 2m1 m2 = 2(โ1) = โ2 Hence slope of lines are ๐/๐ and 1 or 1 and 2 or ( โ๐)/๐ and โ1 or โ1 and โ2
Ex 10.1
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