Introduction / Context:
This question tests understanding of coordinate geometry and section formula concepts. You are asked to find the ratio in which the y-axis divides the line joining two points in the plane. Since both given points lie on the right side of the y-axis, the division is external, and the question explicitly asks you to consider this external division. This type of problem commonly appears in geometry sections of aptitude exams.
Given Data / Assumptions:
- Point A has coordinates (2, 5).
- Point B has coordinates (6, 10).
- The y-axis is the line x = 0.
- We are to find the ratio in which x = 0 divides line AB, considering external division.
Concept / Approach:
For external division, if a point P divides the line segment joining A(x1, y1) and B(x2, y2) externally in the ratio m : n, the x-coordinate of P is given by x = (m x2 - n x1) / (m - n). Since the dividing line is the y-axis (x = 0), we set this x-coordinate to zero and use the relation to determine m : n. For a vertical line x = 0, this leads to a simple relationship between m, n and the x-coordinates of A and B.
Step-by-Step Solution:
1) Let A(2, 5) and B(6, 10).
2) Suppose the y-axis (x = 0) divides the line joining A and B externally at point P in the ratio m : n, where AP : PB = m : n.
3) For external division, the x-coordinate of P is given by x = (m x2 - n x1) / (m - n).
4) Here x1 = 2 and x2 = 6.
5) Since P lies on the y-axis, its x-coordinate is 0, so (m * 6 - n * 2) / (m - n) = 0.
6) The numerator must be zero, so 6m - 2n = 0.
7) Rearranging gives 6m = 2n, or n = 3m.
8) Therefore the ratio m : n = m : 3m = 1 : 3.
Verification / Alternative check:
We can interpret this as the y-axis dividing the line externally so that A is one part away from P and B is three parts away on the line extended. The ratio of distances |AP| : |PB| is therefore 1 : 3, which is what we obtained algebraically. Although we do not need to compute the exact coordinates of P here, the ratio 1 : 3 is consistent with the requirement that P lies on x = 0 when using the external section formula.
Why Other Options Are Wrong:
Option A (3 : 1) is simply the reverse of the correct ratio and would correspond to n : m rather than m : n. Options C (2 : 5), D (5 : 2) and E (4 : 1) do not satisfy the equation 6m = 2n, so they cannot yield an x-coordinate of zero when substituted into the external section formula. Only 1 : 3 makes the numerator 6m - 2n equal to zero for some positive m, n and thus correctly places the division point on the y-axis.
Common Pitfalls:
A common mistake is to apply the internal division formula even when the division is external, which leads to incorrect equations. Another pitfall is to forget that for the line x = 0, the relevant condition is on the x-coordinate of the dividing point. Some students also confuse the order of the ratio (m : n versus n : m), so it is important to stay consistent with the formula used. Carefully identify whether the division is internal or external and substitute coordinates correctly.
Final Answer:
The y-axis divides the line joining (2, 5) and (6, 10) externally in the ratio
1 : 3.
Discussion & Comments