Ex 9.1, 10
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 θ = |(𝑚2 − 𝑚1)/(1 + 𝑚1𝑚2)|
Here tan θ = 1/3
& m2 = 2m1
Putting values
tan θ = |(𝑚2 − 𝑚1)/(1 + 𝑚1𝑚2)|
1/3 = |(2𝑚1 − 𝑚1)/(1 + 𝑚1(2𝑚1))|
1/3 = |𝑚_1/(1 + 2〖𝑚_1〗^2 )|
|𝑚_1/(1 + 2〖𝑚_1〗^2 )| = 1/3
So, 𝑚_1/(1 + 2〖𝑚_1〗^2 ) = 1/3 or 𝑚_1/(1 + 2〖𝑚_1〗^2 ) = ( −1)/3
Solving 𝒎_𝟏/(𝟏 + 𝟐〖𝒎_𝟏〗^𝟐 ) = 𝟏/𝟑
3m1 = 1 + 2〖"m1" 〗^2
2〖"m1" 〗^2 + 1 – 3m1 = 0
2〖"m1" 〗^2 – 3m1 + 1 = 0
2〖"m1" 〗^2 – 2m1 – m1 + 1 = 0
2m1(m1 – 1) – 1(m1 – 1) = 0
(2m1 – 1) (m1 – 1) = 0
So, m1 = 𝟏/𝟐 , m1 = 1
Solving 𝒎_𝟏/(𝟏 + 𝟐〖𝒎_𝟏〗^𝟐 ) = (−𝟏)/𝟑
3m1 = –1 – 2〖"m1" 〗^2
2〖"m1" 〗^2 + 1 + 3m1 = 0
2〖"m1" 〗^2 + 3m1 + 1 = 0
2〖"m1" 〗^2 + 2m1 + m1 + 1 = 0
2m1(m1 + 1) + 1(m1 + 1) = 0
(2m1 + 1) (m1 + 1) = 0
So, m1 = (−𝟏)/𝟐 , m1 = –1
When m1 = ( 𝟏)/𝟐
m2 = 2m1
m2 = 2(1/2) = 1
When m1 = 1
m2 = 2m1
m2 = 2(1) = 2
When m1 = ( −𝟏)/𝟐
m2 = 2m1
m2 = 2(( − 1)/2) = –1
When m1 = –1
m2 = 2m1
m2 = 2(−1) = –2
Hence slope of lines are 𝟏/𝟐 and 1
or 1 and 2
or ( −𝟏)/𝟐 and −1
or −1 and −2

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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