Q.284. Given sides and a diagonal of a poarallelogram, to find the remaining sides.

Question 284.
Two equations of lines which are parallelogram sides are: 3x-10y+37=0, 9x+2y-17=0, and one of its diagonals 3x-2y-19=0. Find the other two lines of the parallelogram.

Answer 284.
As the lines forming the sides of the parallelogram
3x - 10y + 37 = 0 and
9x + 2y - 17 = 0 are not parallel,
they are the adjacent sides of the parallelogram.
Solving them their point of intersection is (1, 4).

As the point (1, 4) does not lie on the given diagonal,
the points of intersection of the diagonal with the sides
contain the other two sides.

Solving the equation of the diagonal 3x - 2y - 19 = 0 with the side 3x - 10y + 37 = 0,
the point of intersection is (11, 7)
The side parallel to 9x + 2y - 17 = 0 is of the form
9x + 2y = k
(11, 7) lies on it
=> 99 + 14 = k
=> one of the side is
9x + 2y = 113 ... (1)

Solving the equation of the diagonal 3x - 2y - 19 = 0 with the side 9x + 2y - 17 = 0,
the point of intersection is (3, - 5)
The side parallel to 3x - 10y + 37 = 0 is of the form
3x - 10y = k
(3, - 5) lies on it
=> k = 59
=> the second side is
3x - 10y = 59 ... ( 2).

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