There are three points P, Q and R on a straight line such that PQ : QR = 3 : 5. If n is the number of possible values of PQ : PR, then what is n equal to ?

Correct Option: (b) 2

Given three points P, Q, and R on a straight line such that PQ : QR = 3 : 5. Since P, Q, and R are collinear, the relative order of the points on the line can be arranged in different ways: Case 1: Q lies between P and R. Then PR = PQ + QR = 3 + 5 = 8. Thus, PQ : PR = 3 : 8. Case 2: P lies between Q and R. Then PR = QR - PQ = 5 - 3 = 2. Thus, PQ : PR = 3 : 2. Case 3: R lies between P and Q. Then PR = PQ + QR is not possible since QR is part of PQ? Wait, if R is between P and Q, then PQ = PR + RQ => 3 = PR + 5, which gives PR = -2 (lengths cannot be negative in this arrangement unless directed distances, but ratio of lengths is positive). Wait, let's check directed distances or standard line segment arrangements. If points are on a line, either Q is between P and R, or P is between Q and R, or R is between P and Q. But wait, if PQ : QR = 3 : 5, let PQ = 3k and QR = 5k. If R is between P and Q, then PQ = PR + RQ => 3k = PR + 5k => PR = -2k, which is geometrically invalid for distance ratio unless signed. However, in standard UPSC coordinate/geometry questions of this type, let's check if the ratio PQ : PR has a unique value or multiple values. Wait! Let's re-verify: Does the relative position matter? If Q is between P and R, PQ/PR = 3/8. If P is between Q and R, PQ/PR = 3/2. Are there 2 possible values? Let's check option (b) which is 2. Yes, n = 2.