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Misc 11 - Lines through (3, 2) which make angle 45 with x-2y=3 - Miscellaneous

  1. Chapter 10 Class 11 Straight Lines
  2. Serial order wise
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Misc 11 Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x – 2y = 3. Let the equation line AB be x − 2y = 3 Let line CD pass through point (3, 2) & make an angle of 45° with line AB i.e. angle between CD & AB is 45° Let m1 be the slope of line AB & m2 be the slope of line CD Given angle between CD & AB is 45° i.e. θ = 45° Now, angle between two lines is given by tan θ =|(𝑚_2 − 𝑚_1)/(1 + 𝑚_2 𝑚_1 )| Here, m1 = Slope of line AB m2 = Slope of line CD & θ = angle between AB & CD = 45° Putting values tan 45° =|(𝑚_2 − 𝑚_1)/(1 + 𝑚_2 𝑚_1 )| 1 = |(𝑚_2 − 𝑚_1)/(1 + 𝑚_2 𝑚_1 )| Finding slope of line AB Given equation of line AB is x − 2y =3 − 2y = 3 − x y = (3 − 𝑥)/( − 2) y = 3/( − 2) + (( − 𝑥)/( − 2)) y = (−3)/2 + 𝑥/3 y = 𝑥/3 + ((−3)/2) y = 1/3 x – 3/2 The above equation is of the form y = mx + c Where m is slope of line Slope of line AB = 1/3 i.e. m1 = 1/3 Putting m1 = 1/3 in (1) 1 = |(𝑚_2 − 1/2)/(1 + 𝑚_2 × 1/2)| 1 = |( (2𝑚_2 − 1)/2)/( (2 + 𝑚_2)/2)| 1 = |(2𝑚_2 − 1)/(2 + 𝑚_2 )| |(2𝑚_2 − 1)/(2 + 𝑚_2 )| = 1 Hence, (2𝑚_2 − 1)/(2 + 𝑚_2 ) = 1 or (2𝑚_2 − 1)/(2 + 𝑚_2 ) = − 1 So, m2 = 3 or ( − 1)/3 ∴ Slope of line CD is 3 or ( − 1)/3 Now we need to calculate equation of line CD Equation of line passes through (x1, y1) & having slope m is (y – y1) = m(x – x1) Equation of line CD passing through (3, 2) & having slope m2 is (y – 2) = m2(x – 3) Hence required equation of line is 3x − y = 7 or x + 3y = 9

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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