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Standard XII

Mathematics

Section Formula

Square ABCD...

Question

Square $A B C D$ is inscribed in a circle, and $P$ is a point on arc $B C$ of the circle.

\frac{P A + P C}{P B + P D}

=

\frac{P D}{P A}

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\frac{P D + P C}{P B + P A}

=

\frac{P D}{P A}

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\frac{P A + P C}{P B + P D}

=

\frac{P A}{P D}

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None of the Above.

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Solution

The correct option is B $\frac{P A + P C}{P B + P D}$ $=$ $\frac{P D}{P A}$
By applying Ptolemy's Theorem to cyclic quadrilaterals $P B A D$ and $P A D C$ respectively we have:
$P A \times B D = P B \times A D + P D \times A B = A B (P B + P D)$
$P D \times A C = P C \times A D + P A \times C D = A B (P A + P C)$
Now by dividing these two we get,
$\frac{P A + P C}{P B + P D}$ $=$ $\frac{P D}{P A}$

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Square ABCD is inscribed in a circle, and P is a point on arc BC of the circle.

The correct option is B PA+PCPB+PD = PDPABy applying Ptolemy's Theorem to cyclic quadrilaterals PBAD and PADC respectively we have:PA×BD=PB×AD+PD×AB=AB(PB+PD)PD×AC=PC×AD+PA×CD=AB(PA+PC)Now by dividing these two we get,PA+PCPB+PD = PDPA

Square $A B C D$ is inscribed in a circle, and $P$ is a point on arc $B C$ of the circle.