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# Is the line through (– 2, 3) and (4, 1) perpendicular to the line 3x = y + 1? Does the line 3x = y + 1 bisect the join of (– 2, 3) and (4, 1).

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An Important Question of class 10 Based on Equation of a Straight Line Chapter of M.L Aggarwal for ICSE BOARD.
Here two co-ordinated are given and prove that it is perpendicular to a given line and does this line bisect the join of given co-ordinated
This is the Question Number 08, Exercise 12.2 of M.L Aggarwal.

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1. Slope of the line passing through the points (-2, 3) and (4, 1) is given by

m= y2 – y1/ x2 – x1

= (1 – 3)/ (4 + 2)

= -2/6

= -1/3

And, the slope of the line: 3x = y + 1

y = 3x -1

Slope (m2) = 3

Now,

m1 x m= -1/3 x 3 = -1

Thus, the lines are perpendicular to each other as the product of their slopes is -1.

Now,

Co-ordinates of the mid-point of the line joining the points (-2, 3) and (4, 1) is

([-2 + 4]/2, [3 + 1]/2) = (1, 2)

Now, if the line 3x = y + 1 passes through the mid-point then it will satisfy the equation

3(1) = (2) + 1

3 = 3

Hence, the line 3x = y + 1 bisects the line joining the points (– 2, 3) and (4, 1).

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