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

 

 ncert cbse chapter 10 straight lines miscellaneous exercise

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. 

let m be the slope of the required line

using formula slope = [(- coefficient of x) / (coefficient of y)]

Slope of  the given line [ x – 2y = 3 ] 

is m2 = [(-1)/(-2)] = [1/2]

using formula for angle between two lines

tanu = | [m1 - m2] / [ 1 + m1*m2]| 

given angle is 45 degrees

tan(45 degrees) = | [m - (1/2)] / [ 1 + m*(1/2)]| 

1  = | [2m-1] / [2+m] |

because of the absolute value sign

(+1) = [2m-1] / [2+m]  OR  (-1) = [2m-1] / [2+m] 

solving

2+m = 2m -1  or  -2-m = 2m-1

m=3  or m=(-1/3)

Required line passes through (3, 2) 

using point slope form

equation is

[y-2] =[3][x-3]  or [y-2] =[(-1/3)][x-3]

y-2 =3x-9   or 3y-6 =(- x)+3

3x-y-7=0  or x+3y-9=0

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

 

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

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

 

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