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If the lines 3x + y = 4, x – ay + 7 = 0 and bx + 2y + 5 = 0 form three consecutive sides of a rectangle, find the value of a and b.

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M.L Aggarwal book Important Question of class 10 chapter Based on Equation of a Straight Line for ICSE BOARD.
If the lines given in the question form three consecutive sides of a rectangle.
Find the value of as asked in question.
This is the Question Number 10, Exercise 12.2 of M.L Aggarwal.

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1 Answer

  1. Given lines are:

    3x + y = 4 … (i)

    x – ay + 7 = 0 … (ii)

    bx + 2y + 5 = 0 … (iii)

    It’s said that these lines form three consecutive sides of a rectangle.

    So,

    Lines (i) and (ii) must be perpendicular

    Also, lines (ii) and (iii) must be perpendicular

    We know that, for two perpendicular lines the product of their slopes will be -1.

    Now,

    Slope of line (i) is

    3x + y = 4 ⇒ y = -3x = 4

    Hence, slope (m1) = -3

    And, slope of line (ii) is

    x – ay + 7 = 0 ⇒ ay = x + 7

    y = (1/a) x + 7/a

    Hence, slope (m2) = 1/a

    Finally, the slope of line (iii) is

    bx + 2y + 5 = 0 ⇒ 2y = -bx – 5

    y = (-b/2) x – 5/2

    Hence, slope (m3) = -b/2

    As lines (i), (ii) and (iii) are consecutive sides of rectangle, we have

    m1 x m2 = -1 and m2 x m3 = -1

    (-3) x (1/a) = -1 and (1/a) x (-b/2) = -1

    -3 = -a and -b/2a = -1

    a = 3 and b = 2a ⇒ b = 2(3) = 6

    Thus, the value of a is 3 and the value of b is 6.

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