1. **State the problem:** We have two lines given by equations: $$2x + ty = -1$$ $$-3x - gy = 2$$ where $t$ and $g$ are constants. The lines are perpendicular. We need to find the value of $-5tg$. 2. **Rewrite each line in slope-intercept form $y = mx + b$ to find their slopes:** For line (d): $$2x + ty = -1 \implies ty = -2x - 1 \implies y = -\frac{2}{t}x - \frac{1}{t}$$ Slope of (d) is $m_d = -\frac{2}{t}$. For line (m): $$-3x - gy = 2 \implies -gy = 3x + 2 \implies y = -\frac{3}{g}x - \frac{2}{g}$$ Slope of (m) is $m_m = -\frac{3}{g}$. 3. **Use the condition for perpendicular lines:** Two lines are perpendicular if the product of their slopes is $-1$: $$m_d \times m_m = -1$$ Substitute the slopes: $$\left(-\frac{2}{t}\right) \times \left(-\frac{3}{g}\right) = -1$$ Simplify: $$\frac{6}{tg} = -1$$ 4. **Solve for $tg$:** $$6 = -tg \implies tg = -6$$ 5. **Find $-5tg$:** $$-5tg = -5 \times (-6) = 30$$ **Final answer:** $-5tg = 30$ which corresponds to option D.