Question

The point P(-4, 6) divides the line segment joining A(-6, 10) and B(3, -8) internally in the ratio

• A. 2 : 7
• B. 7 : 2
• C. 3 : 7
• D. 2 : 5

Hint:

Using only one coordinate (either x or y) is sufficient to find the ratio.
Using the x-coordinate is often simpler as the numbers are smaller and less prone to calculation errors.

Answer 1

The Correct Option is A

Step 1: Use distances instead of the section formula.
Since $P$ lies on segment $AB$, the ratio in which it divides $AB$ is the same as the ratio of the straight line distances $AP : PB$.
Step 2: Find AP.
With $A(-6, 10)$ and $P(-4, 6)$, \[ AP = \sqrt{(-4-(-6))^2 + (6-10)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt5 \]
Step 3: Find PB.
With $P(-4, 6)$ and $B(3, -8)$, \[ PB = \sqrt{(3-(-4))^2 + (-8-6)^2} = \sqrt{7^2 + (-14)^2} = \sqrt{49 + 196} = \sqrt{245} = 7\sqrt5 \]
Step 4: Form the ratio.
\[ AP : PB = 2\sqrt5 : 7\sqrt5 = 2 : 7 \]
\[ \boxed{2 : 7} \]