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Question

# Let any tangent plane to the sphere (xāa)2+(yāb)2+(zāc)2=r2 makes intercepts a,b,c with the coordinate axes at A,B,C respectively. If P is the centre of the sphere, then (ar. and vol. denote the area and volume respectively)

A
vol.(PABC)=abc3
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B
ar.(ABC)=abcr
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C
ar.(PAB)=abcr
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D
vol.(PABC)=abc6
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Solution

## The correct option is C ar.(△PAB)=abcrEquation of the plane is xa+yb+zc=1 Distance of the plane from the centre P(a,b,c) of the sphere is 2√1a2+1b2+1c2=r⇒1a2+1b2+1c2=4r2 Now, −−→AB=−a^i+b^j and −−→AC=−a^i+c^k ar.(△ABC)=12|−−→AB×−−→AC|=12∣∣ ∣ ∣∣∣∣ ∣ ∣∣^i^j^k−ab0−a0c∣∣ ∣ ∣∣∣∣ ∣ ∣∣=12√(ab)2+(bc)2+(ca)2=abc2√1c2+1a2+1b2=abcr −−→PA=−b^j−c^k ar.(△PAB)=12|−−→AB×−−→PA|=12∣∣ ∣ ∣∣∣∣ ∣ ∣∣^i^j^k−ab00−b−c∣∣ ∣ ∣∣∣∣ ∣ ∣∣=12√(ab)2+(bc)2+(ca)2=abc2√1c2+1a2+1b2=abcr vol.(PABC)=13×base area×height =ar.(△ABC)×r3=abc3

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