LINE PASSING THROUGH THE LINE OF INTERSECTION OF THE GIVEN LINES

Line Passing Through Line of Intersection of the Given Lines :

Here we are going to see, how to find the equation of the line passing through line of intersection of the given lines.

Line Passing Through Line of Intersection of the Given Lines - Questions

Question 1 :

Find the equation of a straight line joining the point of intersection of 3x + y + 2 = 0 and x − 2y − 4 = 0 to the point of intersection of 7x − 3y = −12 and 2y = x + 3

Solution :

3x + y + 2 = 0 -----(1)

x − 2y − 4 = 0 -----(2)

2(1) + (2)

6x + 2y + 4  =  0

  x - 2y - 4   =  0

------------------------

7x  =  0

x  =  0/7

x  =  0

By applying the value of x in (1), we get 

3(0) + y + 2  =  0

y  =  -2

Point of intersection of the first two lines is (0, -2)

 7x − 3y = −12 -------(3) 

 2y = x + 3

 x - 2y  =  -3 -------(4)  

2(3) - 3(4)

14x - 6y  =  -24

3x - 6y  =  -9

(-)    (+)   (+)

--------------------

                                     11x  =   -15

x  =  -15/11

By applying x = -15/11 in (4), we get 

(-15/11) - 2y  =  -3

-2y  =  -3 + (15/11)

-2y  =  (-33 + 15)/11

-2y  =  (-33 + 15)/11

-2y  =  -18/11

y = 9/11

Point of intersection of other set of lines is (-15/11, 9/11).

Now, we have to find the equation o the line passing through the points (0, -2) and (-15/11, 9/11).

(y - y₁)/(y₂ - y₁)  =  (x - x₁)/(x₂ - x₁)

(y + 2)/((9/11) + 2)  =  (x - 0)/(-15/11 - 0)

(y + 2)/(31/11)  =  (x - 0)/(-15/11)

-15(y + 2)  =  31(x)

-15y - 30  =  31x

31x + 15y + 30  =  0

Question 2 :

Find the equation of a straight line through the point of intersection of the lines 8x + 3y = 18, 4x + 5y = 9 and bisecting the line segment joining the points (5,–4) and (–7,6).

Solution :

8x + 3y = 18 ----(1)

4x + 5y = 9  ----(2)

5(1) - 3(2)

40x + 15y = 90

12x + 15y  =  27

(-)     (-)     (-)

--------------------

28x  =  63

x  =  63/28

By applying the value of x in (1), we get 

8(63/28) + 3y  =  18

3y  =  18 - (126/7)

3y  =  (126-126)/7

y  =  0 

Point of intersection of the given lines is (63/28, 0). 

(5,–4) and (–7,6)

Midpoint  =  (5 - 7)/2, (-4 + 6)/2

   =  -2/2, 2/2

  =  (-1, 1)

Equation of the line passing through the points (-1, 1) and (63/28, 0)

(y - y₁)/(y₂ - y₁)  =  (x - x₁)/(x₂ - x₁)

(y - 1)/(0 - 1)  =  (x + 1)/((63/28) + 1)

(y - 1)/(- 1)  =  (x + 1)/(91/28)

91(y - 1)  =  -28(x + 1)

91y - 91  =  -28x - 28

28x + 91y - 91 + 28  =  0

28x + 91y - 63  =  0

Dividing the entire equation by 7, we get

4x + 13y - 9  =  0

After having gone through the stuff given above, we hope that the students would have understood, "Line Passing Through Line of Intersection of the Given Lines". 

Apart from the stuff given in this section "Line Passing Through Line of Intersection of the Given Lines", if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.