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:

Shortcut: If a line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) is divided by the y-axis, the ratio is simply \(-x_1 : x_2\). Here, \(-(-4) : 10 = 4 : 10 = 2 : 5\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When a line segment is divided by the y-axis, the x-coordinate of the point of intersection is always \(0\). We use the section formula to find the ratio.
Step 2: Key Formula or Approach:
Let the ratio be \(k:1\). The x-coordinate of the point dividing the line joining \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ x = \frac{k x_2 + x_1}{k + 1} \]
Step 3: Detailed Explanation:
Points are \(P(-4, -2)\) and \(Q(10, 4)\).
Since the division is by the y-axis, the x-coordinate of the point is \(0\).
Let ratio be \(k:1\).
\[ 0 = \frac{k(10) + 1(-4)}{k + 1} \]
\[ 0 = 10k - 4 \]
\[ 10k = 4 \]
\[ k = \frac{4}{10} = \frac{2}{5} \]
The ratio \(k:1\) becomes \(\frac{2}{5}:1\), which is \(2:5\).
Step 4: Final Answer:
The ratio is \(2:5\).