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Question

A straight line through the vertex P of a traingle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then

A
1PS+1ST<1QS×SR
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B
1PS+1ST>2QS×SR
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C
1PS+1ST<4QR
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D
1PS+1ST>4QR
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Solution

The correct option is D 1PS+1ST>4QR

We know by geometry PS×ST=QS×SR ......(1)
S is not the centre of circulm circle,
PSST
And we know that for two unequal real numbers.
H.M < G.M.
21PS+1ST<PS×ST1PS+1ST>2PS×ST
1PS+1ST>2QS×SR [using eqn (1)] ....(2)
(b) is the correct option.
Also QS×SR<QS+SR2(GM<AM)
1QS×SR>2QR2QS×SR>4QR....(3)
From equations (2) and (3) we get 1PS+1ST>4QR
(d) is also the correct option.

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