The set of values that \(\beta\) can assume so that the point \((0, \beta)\) should lie on or inside the…

The set of values that \(\beta\) can assume so that the point \((0, \beta)\) should lie on or inside the triangle having sides \(3 x+y+2=0\), \(2 x-3 y+5=0\) and \(x+4 y-14=0\), is
  1. \(\left[\frac{5}{3}, \frac{7}{2}\right]\)
  2. \(\left[\frac{2}{3}, \frac{5}{2}\right]\)
  3. \(\left[-\frac{1}{3}, \frac{2}{3}\right]\)
  4. \(\left[\frac{1}{2}, \frac{5}{2}\right]\)

Solution

According to given information, the diagram will be as following
Let \(L_1 \equiv 3 x+y+2=0\) \(\begin{aligned} & \Rightarrow \quad \frac{x}{-\frac{2}{3}}+\frac{y}{-2}=1 \\ & \qquad \quad \frac{x}{L_2}+\frac{y}{5}=1 \\ & \Rightarrow \quad \frac{x}{3} \\ & \text {and } \quad \frac{L_3}{14} \equiv x+4 y-14=0 \\ & \Rightarrow \quad \frac{y}{7 / 2}=1 \end{aligned}\) It is clear from the diagram that set of values of \(\beta \in\left[\frac{5}{3}, \frac{7}{2}\right].\)

Asked in: AP EAMCET 2019 (22 Apr Shift 1)

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