If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

slope of y = mx + c is m.

 

slope of y=3x+1 is m1 = 3

 2y = x + 3 can be rewritten as y =(x/2)+(3/2) so slope is m2 = (1/2)

slope of y =mx +4 is m3 = m

angle u between two lines with slopes m1 and m2 is given by

tan(u) = |[m1-m2] / [ 1+m1*m2 ]|

angle u1 between y = mx + c and y=3x+1 is given by

tan(u1) = |[m-3] / [ 1+3m ]|

angle u2 between y = mx + c and 2y = x + 3 is given by

 tan(u1) = |[m-(1/2)] / [ 1+(m/2) ]|

given that  y = mx + c is equally inclined to the other two lines

|[m-3] / [ 1+3m ]| = |[m-(1/2)] / [ 1+(m/2) ]|

 

removing the absolute sign gives two equations

 [m-3] / [ 1+3m ] = +{[m-(1/2)] / [ 1+(m/2) ] }

 [m-3] / [ 1+3m ] = (-){[m-(1/2)] / [ 1+(m/2) ] }

 

first equation changes to 

 [m-3] / [ 1+3m ] = {[2m-1] / [ 2+m] }

  [m-3][ 2+m] =[2m-1][ 1+3m ]

 

(m^2 ) - m  -6 = -1-m +6(m^2)

5(m^2) = (- 5)

(m^2) = (- 1) which is not possible.

second equation changes to

[m-3] / [ 1+3m ] = (-){[2m-1] / [ 2+m] }

  [m-3][ 2+m] =(-)[2m-1][ 1+3m ]

 

(m^2 ) - m  -6 = +1+m - 6(m^2)

7 (m^2 ) - 2m -7 =0

a = 7 b =(-2) c=(-7)

using quadratic formula

m = { (-b )  +   sqrt[(b^2) - 4ac] } / [2a]

or

m = { (-b )  -   sqrt[(b^2) - 4ac} / [2a]

 

m =[2 + 2sqrt(50)] / [14]  or m =[2 - 2sqrt(50)] / [14]

cancelling 2 and factoring

m =  [1 + 5sqrt(2)] / [7]  or m =  [1 - 5sqrt(2)] / [7]

 

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the
line to be a plane mirror.

solution 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

 

ncert cbse chapter 9 sequences and series miscellaneous exercise

 

 32.

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day. 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was finished

solution

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means.