In a ΔXYZ, let x,y,z be the lengths of sides opposite to the angles X,Y,Z respectively and 2s=x+y+z. If s−x4=s−y3=s−z2 and area of incircle of the ΔXYZ is 8π3. then

s−x4=s−y3=s−z2=3s−x+y+z4+3+2=s9=λsay⇒s=9λ,s=4λ+x,s=3λ+y,s=2λ+z∴s=9λ,x=5λ,y=6λ,z=7λΔ=ss−xs−ys−z=66.λ2Also, πr2=8π3⇒r2=83R=xyz4Δ=35λ46r2=83Δ2s2=216λ481λ2⇒83=83λ2⇒λ=1aΔxyz=66.λ2=66bR=35λ46=3546cr=4Rsinx2siny2.sinz2=223=4.3546.sinX2.sinY2.sinZ2⇒435=sinX2sinY2sinZ2dsin2X+Y2=cos2Z2x+y2=90°−z2=ss−zxy=9×2s×6=35

In a ΔXYZ, let x,y,z be the lengths of sides opposite to the angles X,Y,Z respectively and 2s=x+y+z. If s−x4=s−y3=s−z2 and area of incircle of the ΔXYZ is 8π3. then

s−x4=s−y3=s−z2=3s−x+y+z4+3+2=s9=λsay⇒s=9λ,s=4λ+x,s=3λ+y,s=2λ+z∴s=9λ,x=5λ,y=6λ,z=7λΔ=ss−xs−ys−z=66.λ2Also, πr2=8π3⇒r2=83R=xyz4Δ=35λ46r2=83Δ2s2=216λ481λ2⇒83=83λ2⇒λ=1aΔxyz=66.λ2=66bR=35λ46=3546cr=4Rsinx2siny2.sinz2=223=4.3546.sinX2.sinY2.sinZ2⇒435=sinX2sinY2sinZ2dsin2X+Y2=cos2Z2x+y2=90°−z2=ss−zxy=9×2s×6=35

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In a $Δ X Y Z,$ let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$ respectively and $2 s = x + y + z .$ If $\frac{s - x}{4} = \frac{s - y}{3} = \frac{s - z}{2}$ and area of incircle of the $Δ X Y Z$ is $\frac{8 π}{3} .$ then

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area of the $Δ X Y Z$ is $6 \sqrt{6}$

the radius of circumcircle of the $Δ X Y Z$ is $\frac{3 s \sqrt{6}}{6}$

$\sin \frac{X}{2} . \sin \frac{Y}{2} . \sin \frac{Z}{2} = \frac{4}{35}$

$\sin^{2} (\frac{X + Y}{2}) = \frac{3}{5}$

answer is A, B, D.

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Detailed Solution

$\frac{s - x}{4} = \frac{s - y}{3} = \frac{s - z}{2} = \frac{3 s - (x + y + z)}{4 + 3 + 2} = \frac{s}{9} = λ (s a y)$
$\Rightarrow s = 9 λ, s = 4 λ + x, s = 3 λ + y, s = 2 λ + z$
$∴ s = 9 λ, x = 5 λ, y = 6 λ, z = 7 λ$
$Δ = \sqrt{s (s - x) (s - y) (s - z)} = 6 \sqrt{6} . λ^{2}$
Also, $π r^{2} = \frac{8 π}{3} \Rightarrow r^{2} = \frac{8}{3}$
$R = \frac{x y z}{4 Δ} = \frac{35 λ}{4 \sqrt{6}}$
$r^{2} = \frac{8}{3} \frac{Δ^{2}}{s^{2}} = \frac{216 λ^{4}}{81 λ^{2}}$
$\Rightarrow \frac{8}{3} = \frac{8}{3} λ^{2}$
$\Rightarrow λ = 1$
$(a) Δ x y z = 6 \sqrt{6} . λ^{2} = 6 \sqrt{6}$
$(b) R = \frac{35 λ}{4 \sqrt{6}} = \frac{35}{4 \sqrt{6}}$
$(c) r = 4 R \sin \frac{x}{2} \sin \frac{y}{2} . \sin \frac{z}{2}$
$= \frac{2 \sqrt{2}}{\sqrt{3}} = 4. \frac{35}{4 \sqrt{6}} . \sin \frac{X}{2} . \sin \frac{Y}{2} . \sin \frac{Z}{2}$
$\Rightarrow \frac{4}{35} = \sin \frac{X}{2} \sin \frac{Y}{2} \sin \frac{Z}{2}$
$(d) \sin^{2} (\frac{X + Y}{2}) = \cos^{2} (\frac{Z}{2})$
$\frac{x + y}{2} = 90 ° - \frac{z}{2}$
$= \frac{s (s - z)}{x y} = \frac{9 \times 2}{s \times 6} = \frac{3}{5}$