In what ratio is the line segment joining the point (-2, -3) and (3, 7) divided by y-axis ?

Question

Accepted Answer

To determine the ratio in which the line segment joining the points (-2, -3) and (3, 7) is divided by the y-axis, we need to find the point of intersection of the line segment with the y-axis. This point will have an x-coordinate of 0, as all points on the y-axis have x = 0.

Let's use the section formula to find this point. The section formula states that if a point (x, y) divides the line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$, then:

$$ x = \frac{mx_2 + nx_1}{m+n} $$
$$ y = \frac{my_2 + ny_1}{m+n} $$

Since the point lies on the y-axis, x = 0. Therefore, we set up the equation for x:

$$ 0 = \frac{m \cdot 3 + n \cdot (-2)}{m+n} $$

Solving for the ratio $m:n$:

$$ 0 = \frac{3m - 2n}{m+n} $$

$$ 3m = 2n $$

$$ \frac{m}{n} = \frac{2}{3} $$

Thus, the line segment is divided by the y-axis in the ratio 2:3.

Now, let's explain why each of the other options is incorrect:

1. **Option 2 (3:0):** This option suggests that the line segment is divided entirely in favor of one point, which would imply that the point of division is one of the endpoints of the segment. However, the y-axis cannot divide the segment at one of its endpoints, as it must intersect the segment at some point between the endpoints.

2. **Option 3 (-2:3):** A negative ratio is not meaningful in this geometric context, as ratios represent a division of a segment into parts, which cannot be negative.

3. **Option 4 (6:0):** Similar to Option 2, this option implies that the division occurs entirely at one endpoint, which is not possible for a division by the y-axis.

Therefore, the correct answer is Option 1 (2:3), as it accurately represents the division of the line segment by the y-axis.