Name: Angle Between Two Lines
Uploaded: 2019-07-19
Duration: 6 min 13 s
Description: If two lines are perpendicular to each other, their direction vectors are also perpendicular. This means the scalar product of the direction vectors is equal...

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.

Important Questions on The Straight Lines

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Prove that the area of the parallelogram formed by the lines a1x+b1y+c1=0, a1x+b1y+d1=0, a2x+b2y+c2=0, a2x+b2y+d2=0 is d1-c1d2-c2a1b2-a2b1 sq. units . Deduce the condition for these lines to form a rhombus.

Important Questions on The Straight Lines

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