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

To find the ratio in which the line segment joining points \( P(-4, -2) \) and \( Q(10, 4) \) is divided by the y-axis, we follow these steps:

1. First, understand that the y-axis will intersect the line segment at a point where the x-coordinate is zero. We use this property to find the x-intercept of the line.
2. The equation of the line through points \( P(x_1, y_1) = (-4, -2) \) and \( Q(x_2, y_2) = (10, 4) \) is given by the formula for the line connecting two points:

\(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)\)

1. Substituting the values:

\(y + 2 = \frac{4 + 2}{10 + 4} (x + 4)\) 
\(y + 2 = \frac{6}{14}(x + 4)\) 
\(y + 2 = \frac{3}{7}(x + 4)\)

• Setting \( x = 0 \) to find the intersection with the y-axis:

\(y + 2 = \frac{3}{7}(4)\)

\(y + 2 = \frac{12}{7}\)

\(y = \frac{12}{7} - \frac{14}{7}\)

\(y = -\frac{2}{7}\)

1. The intersection point on the y-axis is \( \left(0, -\frac{2}{7}\right) \).
2. Now, calculate the distance of this intersection point from the points \( P \) and \( Q \) using the distance formula. The distance is given by:

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

1. Distance from \( P(-4, -2) \) to intersection \( \left(0, -\frac{2}{7}\right) \):

\(d = \sqrt{(0 + 4)^2 + \left(-\frac{2}{7} + 2\right)^2}\)

\(= \sqrt{16 + \left(\frac{14-2}{7}\right)^2}\)

\(= \sqrt{16 + \left(\frac{12}{7}\right)^2}\)

\(= \sqrt{16 + \frac{144}{49}}\)

\(= \sqrt{\frac{784}{49} + \frac{144}{49}}\)

\(= \sqrt{\frac{928}{49}}\)

1. Distance from \( Q(10, 4) \) to intersection \( \left(0, -\frac{2}{7}\right) \):

\(d = \sqrt{(10 - 0)^2 + \left(4 + \frac{2}{7}\right)^2}\)

\(= \sqrt{100 + \left(\frac{30}{7}\right)^2}\)

\(= \sqrt{100 + \frac{900}{49}}\)

\(= \sqrt{\frac{4900}{49} + \frac{900}{49}}\)

\(= \sqrt{\frac{5800}{49}}\)

1. The ratio of the distances will be: \(\frac{\sqrt{\frac{928}{49}}}{\sqrt{\frac{5800}{49}}} = \frac{\sqrt{928}}{\sqrt{5800}}\).
2. Calculate this ratio to get the required ratio of division: \( \frac{2}{5} \).
3. The line is divided by the y-axis in the ratio \(2:5\).

Therefore, the correct answer is \(2:5\).