Area of parallelogram formed by the lines x + 2y = 5, x + 2y = 15, 3x + y = 10 and 6x + 2y = c is greater than or equal to 1. The complete set of values of c is

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$(- \infty, 21)$

$(- \infty, 19] \cup [21, \infty)$

$(20, \infty)$

$[19, \infty)$

answer is D.

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Detailed Solution

Area of parallelogram formed by line $y = m_{1} x + c_{1}$ , $y = m_{1} x + c_{2}$ , $y = m_{2} x + d_{1}$ , $y = m_{2} x + d_{2}$ is given by
Area of parallelogram $= |\frac{(c_{1} - c_{2}) \cdot (d_{1} - d_{2})}{m_{1} - m_{2}}|$
For the given lines $y = - \frac{x}{2} + \frac{5}{2}; y = - \frac{x}{2} + \frac{15}{2} and y = - 3 x + 10; y = - 3 x + \frac{c}{2}$ ,
Area of parallelogram $= \frac{5 . |\frac{c}{2} - 10|}{\frac{5}{2}} \geq 1$
(Given) $\Rightarrow | c - 20 | \geq 1 \Rightarrow 21 or c \leq 19$