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A straight line through the vertex P of a triangle PQR interse

Question

A straight line through the vertex $P$ of a triangle $PQR$ 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

\frac{1}{PS} + \frac{1}{ST} < \frac{2}{QS \times SR}

\frac{1}{PS} + \frac{1}{ST} > \frac{2}{QS \times SR}

\frac{1}{PS} + \frac{1}{ST} < \frac{4}{QR}

\frac{1}{PS} + \frac{1}{ST} > \frac{4}{QR}

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Elijah Discussed

A straight line through the vertex

P

of a triangle

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

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Text solutionVerified

We have

PS \times ST = QS \times SR

\Rightarrow \frac{\frac{1}{PS} + \frac{1}{ST}}{2} > \frac{1}{PS} \times \frac{1}{ST}

\Rightarrow \frac{1}{PS} + \frac{1}{ST} > \frac{2}{QS \times SR}

Also

\frac{QS + SR}{2} > \frac{2}{QR}

\Rightarrow \frac{1}{PS} + \frac{1}{ST} > \frac{4}{QR}

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Question Text	A straight line through the vertex $P$ of a triangle $PQR$ 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
Answer Type	Text solution:1
Upvotes	151