Finding the Ratio of Division by the Y-Axis

The question asks us to find the ratio in which the y-axis divides the line segment joining two given points. This is a classic problem in coordinate geometry that can be solved using the section formula. The y-axis is the line where the x-coordinate of every point is 0.

Understanding the Section Formula

The section formula helps us find the coordinates of a point that divides a line segment joining two given points in a specific ratio. Let the two points be \(A(x_1, y_1)\) and \(B(x_2, y_2)\), and let a point \(P(x, y)\) divide the line segment AB in the ratio \(m:n\). The coordinates of point P are given by:

\[ P(x, y) = \left( \frac{nx_1 + mx_2}{m+n}, \frac{ny_1 + my_2}{m+n} \right) \]

Applying the Section Formula for Y-Axis Division

We are given the points A(-3, -4) and B(1, 2). The line segment AB is divided by the y-axis. Let the point of division on the y-axis be P. Since P lies on the y-axis, its x-coordinate is 0. Let the y-axis divide the line segment AB in the ratio \(m:n\).

Using the section formula for the x-coordinate:

\[ x = \frac{nx_1 + mx_2}{m+n} \]

Substitute the given values: \(x = 0\), \(x_1 = -3\), \(x_2 = 1\).

\[ 0 = \frac{n(-3) + m(1)}{m+n} \]

Since \(m+n\) cannot be zero (as m and n represent a ratio of lengths and must be positive), we can multiply both sides by \((m+n)\):

\[ 0 \times (m+n) = -3n + m \]

\[ 0 = -3n + m \]

Now, we can solve for the ratio \(m:n\):

\[ m = 3n \]

To express this as a ratio \(m:n\), we can divide both sides by \(n\):

\[ \frac{m}{n} = \frac{3}{1} \]

So, the ratio \(m:n\) is 3:1.

Finding the Point of Division (Optional Check)

We can also find the y-coordinate of the point of division using the ratio \(m:n = 3:1\). Let \(m=3\) and \(n=1\). The y-coordinate is:

\[ y = \frac{ny_1 + my_2}{m+n} \]

Substitute the values: \(n=1\), \(y_1 = -4\), \(m=3\), \(y_2 = 2\), \(m+n = 3+1=4\).

\[ y = \frac{1(-4) + 3(2)}{1+3} = \frac{-4 + 6}{4} = \frac{2}{4} = \frac{1}{2} \]

The point where the y-axis divides the line segment is \((0, 1/2)\).

Conclusion

The y-axis divides the line segment joining the points (-3, -4) and (1, 2) in the ratio 3:1. This result comes directly from applying the section formula and using the fact that the x-coordinate on the y-axis is zero.