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a
$\frac{{\mathrm{P}}_{0}\left({\mathrm{a}}^{3}+{\mathrm{b}}^{3}-{\mathrm{c}}^{3}\right)}{4\left({\mathrm{a}}^{2}+{\mathrm{b}}^{2}+{\mathrm{c}}^{2}\right)}$
b
$\frac{4{\mathrm{P}}_{0}\left({\mathrm{a}}^{3}+{\mathrm{b}}^{3}-{\mathrm{c}}^{3}\right)}{\left({\mathrm{a}}^{2}+{\mathrm{b}}^{2}-{\mathrm{c}}^{2}\right)}$
c
$\frac{{\mathrm{P}}_{0}\left({\mathrm{c}}^{3}-{\mathrm{a}}^{3}-{\mathrm{b}}^{3}\right)}{4\left({\mathrm{a}}^{2}+{\mathrm{b}}^{2}-{\mathrm{c}}^{2}\right)}$
d

$\frac{{\mathrm{P}}_{0}\left({\mathrm{a}}^{3}+{\mathrm{b}}^{3}-{\mathrm{c}}^{3}\right)}{4\left({\mathrm{a}}^{2}-{\mathrm{b}}^{2}-{\mathrm{c}}^{2}\right)}$

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detailed solution

Correct option is C

PV = P1V1 + P2V2

$\left({\mathrm{P}}_{0}+\frac{4\mathrm{T}}{\mathrm{R}}\right)\mathrm{V}=\left({\mathrm{P}}_{0}+\frac{4\mathrm{T}}{{\mathrm{R}}_{1}}\right){\mathrm{V}}_{1}+\left({\mathrm{P}}_{0}+\frac{4\mathrm{T}}{{\mathrm{R}}_{2}}\right){\mathrm{V}}_{2}$

$\left({\mathrm{P}}_{0}+\frac{4\mathrm{T}}{\mathrm{R}}\right)\frac{4}{3}{\mathrm{\pi R}}^{3}=\left({\mathrm{P}}_{0}+\frac{4\mathrm{T}}{{\mathrm{R}}_{1}}\right)\frac{4}{3}{\mathrm{\pi R}}_{1}^{3}+\left({\mathrm{P}}_{0}+\frac{4\mathrm{T}}{{\mathrm{R}}_{2}}\right)\frac{4}{3}{\mathrm{\pi R}}_{2}^{3}$

${\mathrm{P}}_{0}{\mathrm{R}}^{3}+4{\mathrm{TR}}^{2}={\mathrm{P}}_{0}{\mathrm{R}}_{1}^{3}+4{\mathrm{TR}}_{1}^{2}+{\mathrm{P}}_{0}{\mathrm{R}}_{2}^{3}+4{\mathrm{TR}}_{2}^{2}$

${\mathrm{P}}_{0}\left({\mathrm{R}}^{3}-{\mathrm{R}}_{1}^{3}-{\mathrm{R}}_{2}^{3}\right)=4\mathrm{T}\left({\mathrm{R}}_{1}^{2}+{\mathrm{R}}_{2}^{2}-{\mathrm{R}}^{2}\right)$

$\mathrm{T}=\frac{{\mathrm{P}}_{0}\left({\mathrm{c}}^{3}-{\mathrm{a}}^{3}-{\mathrm{b}}^{3}\right)}{4\left({\mathrm{a}}^{2}+{\mathrm{b}}^{2}-{\mathrm{c}}^{2}\right)}$

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