Q88 / 160 · AP EAMCET 2020 (21 Sep Shift 2) · Mathematics

If the length of the intercept made on the line \(y=a x\) by the lines \(y=2\) and \(y=6\) is less than 5

  • A
    \(a \in(-\infty, \infty)\)
  • B
    \(a \in\left(\frac{-4}{3}, \frac{4}{3}\right)\)
  • C
    \(a \in\left(\frac{-3}{4}, \frac{4}{3}\right)\)
  • Correct answer
    D
    \(a < \frac{-4}{3}\) or \(a>\frac{4}{3}\)

Solution

The point of intersection of lines \(y=a x\) and \(y=2\) is \(A\left(\frac{2}{a}, 2\right)\) and of the lines \(y=a x\) and \(y=6\) is \(B\left(\frac{6}{a}, 6\right)\), so \(\begin{aligned} & A B=\sqrt{\left(\frac{6}{a}-\frac{2}{a}\right)^2+(6-2)^2} < 5 \quad \text{(given)} \\ & \Rightarrow \quad \frac{16}{a^2}+16 < 25 \\ & \Rightarrow \quad a^2 > \frac{16}{9} \Rightarrow a \in\left(-\infty,-\frac{4}{3}\right) \cup\left(\frac{4}{3}, \infty\right) \\ \end{aligned}\) (given) Hence, option (d) is correct.

Hence, the correct answer is option: 'D'.

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