Question

The intercepts on the straight line \(y = mx\) by the lines \(y = 2\) and \(y = 6\) is less than 5, then \(m\) belongs to

• A. \(\left(-\frac{4}{3}, \frac{4}{3}\right)\)
• B. \(\left(\frac{4}{3}, \frac{8}{3}\right)\)
• C. \(\left(-\infty, -\frac{4}{3}\right) \cup \left(\frac{4}{3}, \infty\right)\)
• D. \(\left(\frac{4}{3}, \infty\right)\)

Hint:

Distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).

Answer 1

The Correct Option is A

To find the values of \(m\) where the intercepts on the line \(y = mx\) by the lines \(y = 2\) and \(y = 6\) are less than 5, we begin by determining these intercepts.

1. The line \(y = mx\) passes through the origin \((0, 0)\) and the slope of this line is \(m\).
2. The intercepts on the line \(y = mx\) by the lines \(y = 2\) and \(y = 6\) occur where these lines intersect \(y = mx\). Therefore, we solve for \(x\) in each case:
   • From \(y = 2\): \[ 2 = mx \implies x = \frac{2}{m} \]
   • From \(y = 6\): \[ 6 = mx \implies x = \frac{6}{m} \]
3. The distance between these intercept points on the x-axis is given by: \[ \left|\frac{6}{m} - \frac{2}{m}\right| = \left|\frac{6 - 2}{m}\right| = \left|\frac{4}{m}\right| \]
4. Given that this distance must be less than 5, we set up the inequality: \[ \left|\frac{4}{m}\right| < 5 \]
5. This simplifies to: \[ -5 < \frac{4}{m} < 5 \]
6. To remove the fraction, multiply all parts of the inequality by \(m^2\) (assuming \(m \neq 0\)): \[ -5m < 4 < 5m \]
7. This results in two inequalities:
   • \(-5m < 4\):
   • \(4 < 5m\):
8. These inequalities imply that: \[ m \in \left(-\frac{4}{3}, \frac{4}{3}\right) \]

Thus, the range for the value of \(m\) is \(\left(-\frac{4}{3}, \frac{4}{3}\right)\). This matches the provided correct answer option.