Question

The line segment joining the points $P(-4, -2)$ and $Q(10, 4)$ is divided by y-axis in the ratio

• A. $2 : 5$
• B. $1 : 2$
• C. $2 : 1$
• D. $5 : 2$

Hint:

To find the ratio in which the $y$-axis divides the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$, you can use the direct shortcut formula:
\[ \text{Ratio} = -x_1 : x_2 \]
Here, $-(-4) : 10 \implies 4 : 10 = 2 : 5$. This saves valuable time in multiple-choice questions!

Answer 1

The Correct Option is A

Step 1: Find the equation of the line PQ.
The slope of line $PQ$ through $P(-4,-2)$ and $Q(10,4)$ is $m = \frac{4-(-2)}{10-(-4)} = \frac{6}{14} = \frac{3}{7}$. Using point $P$, the line is $y + 2 = \frac{3}{7}(x+4)$.
Step 2: Find where this line meets the y-axis.
On the y-axis, $x = 0$. Substituting: \[ y + 2 = \frac{3}{7}(0+4) = \frac{12}{7} \] \[ y = \frac{12}{7} - 2 = -\frac{2}{7} \] So the line meets the y-axis at $R\left(0, -\frac{2}{7}\right)$.
Step 3: Compare the x-coordinates to get the ratio.
Since $R$ divides $PQ$, the ratio $PR:RQ$ equals the ratio of the x-distances covered: from $x=-4$ to $x=0$ is $4$ units, and from $x=0$ to $x=10$ is $10$ units. So $PR:RQ = 4:10 = 2:5$.
\[ \boxed{2:5} \]