Question

Prove that the area of the parallelogram formed by the lines $a_{1} x + b_{1} y + c_{1} = 0, a_{1} x + b_{1} y + d_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0, a_{2} x + b_{2} y + d_{2} = 0$ is $|\frac{(d_{1} - c_{1}) (d_{2} - c_{2})}{a_{1} b_{2} - a_{2} b_{1}}| sq. units$ .

Deduce the condition for these lines to form a rhombus.

Accepted Answer

The given lines are

$a_{1} x + b_{1} y + c_{1} = 0$ ...(1)

$a_{1} x + b_{1} y + d_{1} = 0$ ..(2)

$a_{2} x + b_{2} y + c_{2} = 0$ ...(3)

$a_{2} x + b_{2} y + d_{2} = 0$ ...(4)

The area of the parallelogram formed by the lines $a_{1} x + b_{1} y + c_{1} = 0, a_{1} x + b_{1} y + d_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$ and $a_{2} x + b_{2} y + d_{2} = 0$ is given below:

Area $= |\frac{(c_{1} - d_{1}) (c_{2} - d_{2})}{|\begin{array}{l} a_{1} & a_{2} \\ b_{1} & b_{2} \end{array}|}|$

$∵ |\begin{array}{l} a_{1} & a_{2} \\ b_{1} & b_{2} \end{array}| = a_{1} b_{2} - a_{2} b_{1}$

$∴$ Area $= |\frac{(c_{1} - d_{1}) (c_{2} - d_{2})}{a_{1} b_{2} - a_{2} b_{1}}| = |\frac{(d_{1} - c_{1}) (d_{2} - c_{2})}{a_{1} b_{2} - a_{2} b_{1}}|$

If the given parallelogram is a rhombus, then the distance between the pair of parallel lines are equal.

The distance between the parallel lines $a_{1} x + b_{1} y + c_{1} = 0$ , and $a_{1} x + b_{1} y + d_{1} = 0$ is

$|\frac{c_{1} - d_{1}}{\sqrt{{a_{1}}^{2} + {b_{1}}^{2}}}|$

And, the distance between the parallel lines $a_{2} x + b_{2} y + c_{2} = 0$ , and $a_{2} x + b_{2} y + d_{2} = 0$ is

$|\frac{c_{2} - d_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}|$

$∴ |\frac{c_{1} - d_{1}}{\sqrt{a_{1}^{2} + {b_{1}}^{2}}}| = |\frac{c_{2} - d_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}|$ is the required condition for these lines to form a rhombus.

R D Sharma Solutions for Chapter: The Straight Lines, Exercise 17: EXERCISE 23.17

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: The Straight Lines, Exercise 17: EXERCISE 23.17

Questions from R D Sharma Solutions for Chapter: The Straight Lines, Exercise 17: EXERCISE 23.17 with Hints & Solutions

Learn and Prepare for any exam you want !