Find the Equation of Perpendicular Bisector :
Here we are going to see how to find the equation of perpendicular bisector.
Question 10 :
Find the equation of the perpendicular bisector of the straight line segment joining the points (3,4) and (-1,2)
Solution :
midpoint of the line segment joining the points (3,4) and (-1,2)
x₁ = 3, y₁ = 4, x₂ = -1 , y₂ = 2
= [(x₁ + x₂)/2 , (y₁ + y₂)/2]
= (3 + (-1))/2 , (4 + 2)/2
= (2/2, 6/2)
= (1 , 3)
slope of the line joining the points (3,4) and (-1,2)
m = (y₂ - y₁)/(x₂ - x₁)
= (2 - 4)/(-1-3)
= -2/(-4)
= 1/2
Slope of the required line = -1/(1/2)
= -2
Equation of the required line :
(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
(y - 4)/(2 - 4) = (x - 3)/(-1 - 3)
(y - 4)/(-2) = (x - 3)/(-4)
-4(y - 4) = -2(x - 3)
- 4 y + 16 = - 2 x + 6
2 x -4 y + 16 - 6 = 0
2 x - 4 y + 10 = 0
dividing the whole equation by 2, we get
x - 2 y - 5 =0
Question 11 :
Find the equation of the straight line passing through the point of intersection of the lines 2x+y-3=0 and 5x + y - 6 = 0 and parallel to the line joining the points (1,2) and (2,1)
Solution :
To find the point of intersection of any two lines we need to solve them
2x + y - 3 = 0 ---- (1)
5x + y - 6 = 0 ---- (2)
(1) - (2)
2x + y - 3 = 0
5x + y - 6 = 0
(-) (-) (+)
-----------------
- 3x + 3 = 0
- 3 x = -3
x = 1
Substitute x = 1 in the first equation
2(1) + y - 3 = 0
-1 + y = 0
y = 1
the point of intersection is (1,1)
The required line is parallel to the line joining the points (1, 2) and (2, 1). So their slopes will be equal.
m = (y₂ - y₁)/(x₂ - x₁)
= (1-2)/(2-1)
= -1/1 = -1
Slope of the required line is -1 and a point on the line is (1, 1)
Equation of the Line :
(y - y₁) = m (x - x₁)
(y - 1) = -1 (x - 1)
y - 1 = - x + 1
x + y - 1 - 1 = 0
x + y - 2 = 0
After having gone through the stuff given above, we hope that the students would have understood, how to find the equation of perpendicular bisector.
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