EQUATION OF LINE PASSING THROUGH POINT OF INTERSECTION OF TWO LINES

Equation of Straight Line Passing Through Point of Intersection of Two Lines :

Here we are going to see how to find the equation of the straight line passing through the intersection of two lines and parallel or perpendicular to the given line.

Equation of Line Passing Through Point of Intersection of Two Lines

Question 12 :

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

Find the Parametric Equation of the Line of Intersection of Two Lines

Question 13 :

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

After having gone through the stuff given above, we hope that the students would have understood,how to find equation of line passing through point of intersection of two lines

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