(a) Consider a plane passing through three points A ( a , 0 , 0 ) , B ( 0 , b , 0 ) , C ( 0 , 0 , c ) with a 0 , b 0, c 0 . Let d be the distance between the origin O and the plane and m be the

Question

(a) Consider a plane passing through three points $A (a, 0, 0), B (0, b, 0), C (0, 0, c)$ with $a > 0, b$ > 0, $c > 0$ . Let d be the distance between the origin O and the plane and m be the distance between the origin O and the point M(a,b,c). If a,b,c vary in the range of any positive numbers, then find the minimum value of $(\frac{m}{d})^{2}$ .

(b) Let $A_{1}, A_{2}, A_{3}, A_{4}$ be the areas of the triangular faces of a tetrahedron and $h_{1}, h_{2}, h_{3}, h_{4}$ be corresponding altitudes of the tetrahedron. If volume of tetrahedron is 5 cubic units then find the minimum value of (sum of the four face areas) times (sum of their four altitudes) (in cubic units).

TestHub · Accepted Answer

Apply Cauchy-Schwarz in both parts. In (a), equality occurs when all three intercepts are equal. In (b), a regular tetrahedron attains equality. $(a) \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1, d^{- 2} = a^{- 2} + b^{- 2} + c^{- 2}, m^{2} = a^{2} + b^{2} + c^{2} . (\frac{m}{d})^{2} = (a^{2} + b^{2} + c^{2}) \cdot (a^{- 2} + b^{- 2} + c^{- 2}) \geq (1 + 1 + 1)^{2} = 9. (b) A_{i} h_{i} = 3 V = 15, (i = 1 \sum 4 A_{i}) (i = 1 \sum 4 h_{i}) = 15 (i = 1 \sum 4 A_{i}) (i = 1 \sum 4 \frac{1}{A _{i}}) \geq 15 \cdot 4^{2} = 240.$

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

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