Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# 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 - Mathematics

Short Note

Prove that the area of the parallelogram formed by the lines a1x + b1y + c1 = 0, a1x + b1yd1 = 0, a2x + b2y + c2 = 0, a2x + b2y + d2 = 0 is  $\left| \frac{\left( d_1 - c_1 \right)\left( d_2 - c_2 \right)}{a_1 b_2 - a_2 b_1} \right|$ sq. units.
Deduce the condition for these lines to form a rhombus.

#### Solution

The given lines are
a1x + b1y + c1 = 0      ... (1)
a1x + b1y + d1 = 0      ... (2)
a2x + b2y + c2 = 0      ... (3)
a2x + b2y + d2 = 0      ... (4)
The area of the parallelogram formed by the lines a1x + b1y + c1 = 0, a1x + b1y + d1 = 0, a2x + b2y + c2 = 0 and a2x + b2y + d2 = 0 is given below:

$Area = \left| \frac{\left( c_1 - d_1 \right)\left( c_2 - d_2 \right)}{\begin{vmatrix}a_1 & a_2 \\ b_1 & b_2\end{vmatrix}} \right|$
$\because \begin{vmatrix}a_1 & a_2 \\ b_1 & b_2\end{vmatrix} = a_1 b_2 - a_2 b_1$
$\therefore Area = \left| \frac{\left( c_1 - d_1 \right)\left( c_2 - d_2 \right)}{a_1 b_2 - a_2 b_1} \right| = \left| \frac{\left( d_1 - c_1 \right)\left( d_2 - c_2 \right)}{a_1 b_2 - a_2 b_1} \right|$
If the given parallelogram is a rhombus, then the distance between the pair of parallel lines are equal.
$\therefore \left| \frac{c_1 - d_1}{\sqrt{{a_1}^2 + {b_1}^2}} \right| = \left| \frac{c_2 - d_2}{\sqrt{{a_2}^2 + {b_2}^2}} \right|$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.17 | Q 1 | Page 117