In what ratio does the Y axis (x = 0) divide the line segment joining the points (−1, 3) and (2, −4) in the coordinate plane?

Introduction / Context:
This coordinate geometry problem asks for the ratio in which the Y axis divides a line segment between two given points. It is another application of the section formula or parametric representation combined with the distance formula, and it tests spatial reasoning and algebraic manipulation in the plane.

Given Data / Assumptions:

• The endpoints of the segment are A = (−1, 3) and B = (2, −4).• The Y axis is the line x = 0.• The segment AB intersects the Y axis at some point P.• We must determine the ratio AP : PB.

Concept / Approach:
We can parametrize the segment from A to B, find the parameter value at which x becomes zero, and thus obtain the coordinates of the intersection point P. Once P is known, we use the distance formula to compute distances AP and PB. The ratio of these distances, simplified to whole numbers, gives the required division ratio. This method ensures a clear and systematic solution.

Step-by-Step Solution:
Step 1: Let A = (−1, 3) and B = (2, −4).Step 2: Find the direction vector B − A = (2 − (−1), −4 − 3) = (3, −7).Step 3: Parametrize the line segment as A + t(B − A) for 0 ≤ t ≤ 1.Step 4: A general point on the segment is (−1 + 3t, 3 − 7t).Step 5: The Y axis has equation x = 0, so set the x coordinate equal to zero: −1 + 3t = 0.Step 6: Solve for t: 3t = 1, so t = 1 / 3.Step 7: Substitute t = 1 / 3 into the coordinates to get P: x = −1 + 3 * (1 / 3) = 0 and y = 3 − 7 * (1 / 3) = 3 − 7 / 3 = (9 − 7) / 3 = 2 / 3.Step 8: So P = (0, 2 / 3).Step 9: Now compute distance AP using A(−1, 3) and P(0, 2 / 3).Step 10: AP = sqrt((0 − (−1))^2 + (2 / 3 − 3)^2) = sqrt(1^2 + (2 / 3 − 9 / 3)^2) = sqrt(1 + (−7 / 3)^2) = sqrt(1 + 49 / 9) = sqrt(58 / 9).Step 11: Next, compute distance PB using P(0, 2 / 3) and B(2, −4).Step 12: PB = sqrt((2 − 0)^2 + (−4 − 2 / 3)^2) = sqrt(2^2 + (−14 / 3)^2) = sqrt(4 + 196 / 9) = sqrt(232 / 9).Step 13: Observe that PB^2 / AP^2 = (232 / 9) / (58 / 9) = 232 / 58 = 4.Step 14: Therefore PB^2 = 4 * AP^2 and PB = 2 * AP (taking positive distances).Step 15: So the ratio AP : PB is 1 : 2.

Verification / Alternative check:
The parameter t = 1 / 3 also provides a quick verification. In a parametric representation A + t(B − A), t represents the fraction of the distance from A to B. If t = 1 / 3, then AP : PB = 1 : 2 because AP is one third of the entire segment and PB is the remaining two thirds. This simple interpretation confirms the more detailed distance based calculation and assures us that the ratio AP : PB is indeed 1 : 2.

Why Other Options Are Wrong:
• 2 : 1: This would suggest that AP is longer than PB, which contradicts the finding that PB = 2 * AP.• 1 : 4 and 4 : 1: Both of these imply a much more uneven split of the segment than what the parametric value t = 1 / 3 and distance calculations show.

Common Pitfalls:
Students sometimes confuse which ratio is being asked (AP : PB versus PB : AP) and may invert the correct ratio. Miscomputing the direction vector or the parameter t can also cause errors. Additionally, some learners skip using a parametric form and try to guess the intersection point, leading to algebraic inconsistencies. A structured approach using the equation for the segment and the line x = 0 avoids these issues.

Final Answer:
The Y axis divides the segment joining (−1, 3) and (2, −4) in the ratio 1 : 2 (that is, AP : PB = 1 : 2).