A plane passing through $$1$$,$$1$$,$$1$$ intersects the positive direction of coordinate axes at A,B and C then the volume of the tetrahedron OABC satisfies

Question

A plane passing through $$1$$,$$1$$,$$1$$ intersects the positive direction of coordinate axes at A,B and C then the volume of the tetrahedron OABC satisfies

Infinity Learn · Accepted Answer

Let the equation of the plane in intercept form be $$xa+yb+zc=1$$…………. (1)  Where a,b,c are positive intercepts on coordinate axes.  Eq. (1)_$$ $$) is passing through (1$$,$$1$$,$$1)⇒$$1a+1b+1c=1$$ We know that Volume of the tetrahedron $$OABC=V=16(abc)$$ Using the inequality Geometric mean (G$$.M.$$) ≥ Harmonic mean (H$$.M.$$) ⇒(abc)13$$≥$$31a+1b+1c$$⇒$$(abc)13$$≥$$31$$⇒abc≥$$27$$⇒$$16(abc)≥$$16(27)$$⇒V≥$$92Therefore$$, the correct answer is (2).

A plane passing through $(1, 1, 1)$ intersects the positive direction of coordinate axes at $A, B$ and $C$ then the volume of the tetrahedron $O A B C$ satisfies

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A plane passing through 1,1,1 intersects the positive direction of coordinate axes at A,B and C then the volume of the tetrahedron OABC satisfies

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A plane passing through $(1, 1, 1)$ intersects the positive direction of coordinate axes at $A, B$ and $C$ then the volume of the tetrahedron $O A B C$ satisfies