Discussion

Explanation:

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As S is not the circumcentre, PS ≠ ST and QS ≠ SR

∵ PT and QR are chords of the circle intersecting at S, PS × ST = QS × SR … (i)

We know that Arithmetic mean ≥ Geometric mean

PS+ST2 ≥ PS×ST

But as PS ≠ ST,

PS+ST2 > PS×ST

PS+ST2 > QS×SR

PS+ST2 > 2QS×SR

PS+STPS×ST > 2QS+SRQS×SR

1PS + 1ST > 2QS×SR      ...(i)

∴ Option 1 is false.

Also,

QS+SR2 > QS×SR

2QR < 1QS×SR

4QR < 2QS×SR

1PS + 1ST > 2QS×SR > 4QR    ...From (i)

Hence, option (d).

Note: As the result is a general one, we can, without loss of generality, consider an equilateral triangle PQR with point S being the mid-point of QR and verify all options using numbers.

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