Find the value of t, if the distance between the points P(–3, –14) and Q(t, –5) is 9 units.

Analysis

To find the value of $t$, we need to use the Distance Formula, which calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a Cartesian plane.

Concept

The distance $d$ between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by the formula:

In this problem:

• $P = (-3, -14) \implies x_1 = -3, y_1 = -14$
• $Q = (t, -5) \implies x_2 = t, y_2 = -5$
• $d = 9$

Step-by-Step Solution

1. Substitute the values into the formula: $$9 = \sqrt{(t - (-3))^2 + (-5 - (-14))^2}$$ $$9 = \sqrt{(t + 3)^2 + (-5 + 14)^2}$$

2. Simplify the expression: $$9 = \sqrt{(t + 3)^2 + (9)^2}$$ $$9 = \sqrt{(t + 3)^2 + 81}$$

3. Square both sides to remove the square root: $$9^2 = (t + 3)^2 + 81$$ $$81 = (t + 3)^2 + 81$$

4. Solve for $t$: Subtract 81 from both sides: $$81 - 81 = (t + 3)^2$$ $$0 = (t + 3)^2$$

Take the square root of both sides:

Conclusion

The value of $t$ is $-3$. Comparing this to the given options, it matches option C.

Exam Tip

Whenever you see a distance problem involving a square root, always square both sides immediately to eliminate the root. Also, remember that $(a - (-b))$ becomes $(a + b)$; watch your signs carefully, as a simple sign error is the most common reason students lose marks in coordinate geometry!