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Question

Let any tangent plane to the sphere (xa)2+(yb)2+(zc)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)=abcr
Equation of the plane is xa+yb+zc=1
Distance of the plane from the centre P(a,b,c) of the sphere is
21a2+1b2+1c2=r1a2+1b2+1c2=4r2

Now, AB=a^i+b^j and AC=a^i+c^k
ar.(ABC)=12|AB×AC|=12∣ ∣ ∣∣ ∣ ∣^i^j^kab0a0c∣ ∣ ∣∣ ∣ ∣=12(ab)2+(bc)2+(ca)2=abc21c2+1a2+1b2=abcr

PA=b^jc^k
ar.(PAB)=12|AB×PA|=12∣ ∣ ∣∣ ∣ ∣^i^j^kab00bc∣ ∣ ∣∣ ∣ ∣=12(ab)2+(bc)2+(ca)2=abc21c2+1a2+1b2=abcr

vol.(PABC)=13×base area×height =ar.(ABC)×r3=abc3

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