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(D) The distance between the parallel lines $a_1x + b_1y + c_1 = 0$ and $a_1x + b_1y + d_1 = 0$ is $p_1 = \frac{|d_1 - c_1|}{\sqrt{a_1^2 + b_1^2}}$.Th

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$\frac{|(d_1 - c_1)(d_2 - c_2)|}{|a_1b_2 - a_2b_1|}$

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(D) The distance between the parallel lines $a_1x + b_1y + c_1 = 0$ and $a_1x + b_1y + d_1 = 0$ is $p_1 = \frac{|d_1 - c_1|}{\sqrt{a_1^2 + b_1^2}}$.The distance between the parallel lines $a_2x + b_2y + c_2 = 0$ and $a_2x + b_2y + d_2 = 0$ is $p_2 = \frac{|d_2 - c_2|}{\sqrt{a_2^2 + b_2^2}}$.The area of a parallelogram formed by two pairs of parallel lines is given by $	ext{Area} = \frac{p_1 p_2}{\sin 	heta}$,where $	heta$ is the angle between the lines.The angle $	heta$ between the lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ satisfies $\sin 	heta = \frac{|a_1b_2 - a_2b_1|}{\sqrt{a_1^2 + b_1^2} \sqrt{a_2^2 + b_2^2}}$.Substituting $p_1$,$p_2$,and $\sin 	heta$ into the area formula:$	ext{Area} = \frac{|d_1 - c_1|}{\sqrt{a_1^2 + b_1^2}} 	imes \frac{|d_2 - c_2|}{\sqrt{a_2^2 + b_2^2}} 	imes \frac{\sqrt{a_1^2 + b_1^2} \sqrt{a_2^2 + b_2^2}}{|a_1b_2 - a_2b_1|} = \frac{|(d_1 - c_1)(d_2 - c_2)|}{|a_1b_2 - a_2b_1|}$.

The area of the parallelogram formed by the lines $a_1x + b_1y + c_1 = 0$,$a_1x + b_1y + d_1 = 0$,$a_2x + b_2y + c_2 = 0$,and $a_2x + b_2y + d_2 = 0$ is:

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