A straight line through the vertex P of a - PQR intersects the side QR at the point S and the circum circle of - PQR at the point T- If S is not the centre of the circum circle- then

Given: A straight line through the vertex P of the Δ PQR nbsp;intersects the side QR at the point S and its circum circle at the point T. Hence, P,Q,R,T are c ...

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Standard XII

Mathematics

AM,GM,HM Inequality

A straight li...

Question

A straight line through the vertex $P$ of a $Δ P Q R$ intersects the side $Q R$ at the point $S$ and the circum circle of $Δ P Q R$ at the point $T$ . If $S$ is not the centre of the circum circle, then

\frac{1}{P S} + \frac{1}{S T} < \frac{2}{\sqrt{Q S \times S R}}

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\frac{1}{P S} + \frac{1}{S T} > \frac{2}{\sqrt{Q S \times S R}}

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\frac{1}{P S} + \frac{1}{S T} < \frac{4}{Q R}

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\frac{1}{P S} + \frac{1}{S T} > \frac{4}{Q R}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

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Solution

The correct option is D $\frac{1}{P S} + \frac{1}{S T} > \frac{4}{Q R}$
Given: A straight line through the vertex $P$ of the $Δ P Q R$ intersects the side $Q R$ at the point $S$ and its circum circle at the point $T$ .
Hence, $P, Q, R, T$ are concyclic and thus, $P S . S T = Q S . S R$

Also, using $A M > G M, \frac{P S + S T}{2} > \sqrt{P S . S T}$
Also, $G M > H M \Rightarrow \sqrt{P S . S T} > \frac{2}{\frac{1}{P S} + \frac{1}{S T}} \Rightarrow \frac{1}{P S} + \frac{1}{S T} > \frac{2}{\sqrt{P S . S T}} = \frac{2}{\sqrt{S Q . S R}}$
Also, $\frac{S Q + Q R}{2} > \sqrt{S Q . S R} \Rightarrow \frac{Q R}{2} > \sqrt{S Q . S R} \Rightarrow \frac{1}{\sqrt{S Q . S R}} > \frac{2}{Q R} \Rightarrow \frac{2}{\sqrt{S Q . S R}} > \frac{4}{Q R}$
$∴ \frac{1}{P S} + \frac{1}{S T} > \frac{2}{\sqrt{Q S \times S R}} > \frac{4}{Q R}$

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A straight line through the vertex P of a ΔPQR intersects the side QR at the point S and the circum circle of ΔPQR at the point T. If S is not the centre of the circum circle, then

A straight line through the vertex $P$ of a $Δ P Q R$ intersects the side $Q R$ at the point $S$ and the circum circle of $Δ P Q R$ at the point $T$ . If $S$ is not the centre of the circum circle, then