If the points $(1,2)$ and $(3,4)$ lie on the same side of the straight line $3 x-5 y+a=0$, then $a$ lies in…

If the points $(1,2)$ and $(3,4)$ lie on the same side of the straight line $3 x-5 y+a=0$, then $a$ lies in the set
  1. $[7,11]$
  2. $\mathbb{R}-[7,11]$
  3. $[7, \infty)$
  4. $(-\infty, 11]$

Solution

Since, the points $(1,2)$ and $(3,4)$ lie on the same side of the line $3 x-5 y+a=0$ $ \begin{array}{lrrlrl} & \therefore & 3(1)-5(2)+a & \geq 0 & \text { or } & \leq 0 \\ \Rightarrow & a-7 & \geq 0 & \text { or } \leq 0 \\ \Rightarrow & a & \geq 7 & \text { or } & a \leq 7 \\ \text { and } & & 3(3)-5(4)+a & \geq 0 & \text { or } \leq 0 \\ \Rightarrow & a-11 & \geq 0 & \text { or } \leq 0 \\ \Rightarrow & & a & \geq 11 \text { or } & a \leq 11 \end{array} $ So, common condition is $[7,11]$

Asked in: AP EAMCET 2013

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