In what ratio does the Y axis (the line x = 0) divide the line segment joining the points (2, 3) and (−2, 1) in the coordinate plane?

Introduction / Context:
This coordinate geometry question asks about the ratio in which a vertical line, namely the Y axis (x = 0), divides a segment connecting two given points. It is a standard type of problem that tests understanding of the distance formula and how partition points lie on line segments in the Cartesian plane.

Given Data / Assumptions:

• The two endpoints of the segment are (2, 3) and (−2, 1).• The Y axis is the vertical line given by the equation x = 0.• The segment joining the points intersects the Y axis at some point.• We must find the ratio in which this intersection point divides the segment, measured from one endpoint to the other.

Concept / Approach:
The segment intersects the Y axis where x = 0. We can use a parametric representation of the line segment from one endpoint to the other and then solve for the parameter value at which x equals zero. After getting the intersection point, we compute the distances from this point to each endpoint using the distance formula. The ratio of these two distances gives the required partition ratio. Because distances are always positive, the ratio should be simplified to positive integers.

Step-by-Step Solution:
Step 1: Let A = (2, 3) and B = (−2, 1).Step 2: Parametrize the line segment from A to B as A + t(B − A) for 0 ≤ t ≤ 1.Step 3: Compute B − A = (−2 − 2, 1 − 3) = (−4, −2).Step 4: So a general point on the segment is (2 + t * (−4), 3 + t * (−2)) = (2 − 4t, 3 − 2t).Step 5: The Y axis is x = 0. Set the x coordinate equal to zero: 2 − 4t = 0.Step 6: Solve for t: 4t = 2, so t = 1 / 2.Step 7: Substitute t = 1 / 2 into the coordinates: x = 2 − 4 * 1 / 2 = 0 and y = 3 − 2 * 1 / 2 = 3 − 1 = 2.Step 8: Thus, the Y axis meets the segment at the point P = (0, 2).Step 9: Now compute distances AP and PB using the distance formula.Step 10: AP = sqrt((0 − 2)^2 + (2 − 3)^2) = sqrt((−2)^2 + (−1)^2) = sqrt(4 + 1) = sqrt(5).Step 11: PB = sqrt((−2 − 0)^2 + (1 − 2)^2) = sqrt((−2)^2 + (−1)^2) = sqrt(4 + 1) = sqrt(5).Step 12: Therefore, AP and PB are equal, so the ratio AP : PB is sqrt(5) : sqrt(5) = 1 : 1.

Verification / Alternative check:
The fact that t = 1 / 2 in the parametric form also indicates that the point P lies exactly halfway between A and B. When a point corresponds to t = 1 / 2 on a linear interpolation between two endpoints, it is the midpoint, and by definition the midpoint divides the segment into two equal parts. Hence the ratio of the distances from A to P and from P to B must be 1 : 1. This gives an algebraic confirmation of the earlier distance calculation.

Why Other Options Are Wrong:
• 1 : 2 and 2 : 3: These ratios suggest that the point is closer to one endpoint than the other, which contradicts the equal distances AP and PB.• 3 : 1: This would imply a much greater distance on one side, not supported by the numerical distance calculations.

Common Pitfalls:
Students sometimes do not parametrize correctly or confuse the direction vector B − A with A − B, though this does not affect the final ratio if handled consistently. Another common error is miscalculating the intersection point by solving the equation for the wrong coordinate or misusing the distance formula. Some learners might also forget to simplify the ratio of distances to its simplest integer form. Clearly following the steps of finding the intersection, computing both distances, and then simplifying the ratio prevents these issues.

Final Answer:
The Y axis divides the segment joining (2, 3) and (−2, 1) in the ratio 1 : 1.