EQUATION OF LINE PASSING THROUGH POINT OF INTERSECTION OF TWO LINES

Example 1 :

Find the equation of the straight line which passes through the point of intersection of the straight lines 5x - 6y = 1 and 3x + 2y + 5 = 0 and is perpendicular to the straight line 3x - 5y + 11 = 0.

Solution :

To find the point of intersection of any two lines, we need to solve them

  5x - 6y = 1  --(1)

  3x + 2y = -5 ----(2)

                       5x - 6y = 1

 (2) x 3 =>   9x + 6y = -15

                 ----------------

                  14 x = -14

                    x = -1

Substitute x = -1 in the first equation

  5 (-1) -6 y = 1

-5 - 6y = 1

-6 y = 1 + 5

-6y = 6

y = -1

Point of intersection of those two lines is (-1,-1)

Slope of the perpendicular line 3x - 5y + 11 = 0

m = -3/(-5)

= 3/5

Slope of the required line = -1/m

  = -1/(3/5)

  = -5/3

Equation of the line :

(y - y₁) = m (x - x₁)

(y - (-1)) = (-5/3) (x - (-1))

  3 (y + 1) = -5 (x + 1)

 3 y + 3 = - 5 x - 5

 5 x + 3 y + 5 + 3 = 0 

 5 x + 3y + 8 = 0

Example 2 :

Find the equation of the straight line joining the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4 and the point intersection of the lines 2x + y – 4 = 0 and x – 2y + 3 = 0

Solution :

To find the point of intersection of any two lines we need to solve them

3x – y + 9 = 0 ---- (1)

x + 2y - 4 = 0 ---- (2)

(1) - (2)

              3x – y + 9 = 0

(2) x 3   3x + 6y - 12 = 0

             (-)  (-)  (+)

           ----------------- 

              - 7y + 21 = 0

                    7y = 21, y = 3

Substitute y = 3 in the first equation

3 x - 3 + 9 = 0

3 x + 6 = 0

3 x = - 6

x = -6/3 = -2 

the point of intersection is (-2,3)

2x + y – 4 = 0 ---- (3)

x – 2y + 3 = 0 ---- (4)

(1) - (2)

              2x + y - 4 = 0

(2) x 2   2x - 4y + 6 = 0

             (-)    (+)  (-)

           ----------------- 

                  5 y - 10 = 0

                    5y = 10

                     y = 2

Substitute y = 2 in the third equation

2 x + y - 4 = 0

2 x + 2 - 4 = 0

2 x - 2 = 0

2 x = 2

x = 1

the point of intersection is (1,2)

Equation of the line:

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

(-2,3) (1,2)

x₁ = -2, y₁ = 3, x₂ = 1 , y₂ = 2

(y -3)/(2-3) = (x-(-2))/(1-(-2))

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

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

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

3 y - 9 = - x - 2

x + 3 y - 9 + 2 = 0

x + 3y - 7 = 0

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