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: Understanding the Question:
We are given two points $P(-4, -2)$ and $Q(10, 4)$ forming a line segment.
This line segment is intersected by the $y$-axis.
We need to determine the ratio in which the $y$-axis divides this line segment.
Any point lying on the $y$-axis has its $x$-coordinate equal to zero.

Step 2: Key Formula or Approach:
According to the section formula, if a point $R(x, y)$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $k : 1$, then the coordinates of $R$ are:
\[ x = \frac{kx_2 + x_1}{k+1}, \quad y = \frac{ky_2 + y_1}{k+1} \]
Since the point of division lies on the $y$-axis, its $x$-coordinate is $0$.
We can set the expression for $x$ equal to zero to find the value of $k$.

Step 3: Detailed Explanation:

• Let the coordinates of the given points be:
\[ P(x_1, y_1) = (-4, -2) \]
\[ Q(x_2, y_2) = (10, 4) \]

• Let the $y$-axis divide the line segment $PQ$ at point $R(0, y)$ in the ratio $k : 1$.

• Apply the section formula for the $x$-coordinate of point $R$:
\[ x = \frac{k(10) + 1(-4)}{k+1} \]

• Since $R$ lies on the $y$-axis, its $x$-coordinate is $0$:
\[ 0 = \frac{10k - 4}{k+1} \]

• Cross-multiply to solve for $k$:
\[ 10k - 4 = 0 \]
\[ 10k = 4 \]
\[ k = \frac{4}{10} = \frac{2}{5} \]

• The ratio $k : 1$ is $\frac{2}{5} : 1$, which simplifies to $2 : 5$.
Since $k$ is positive, the division is internal.


Step 4: Final Answer:
The line segment joining the points $P$ and $Q$ is divided by the $y$-axis in the ratio $2 : 5$.