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Most Important Selected Qs for JEE AdvancedMathematicsThree Dimensional Geometry

R and r are the circum-radius and in-radius of a regular tetrahedron respectively in terms of the length k of each edge. If R^2+r^2= p q k^2 , where p, q I then absolute minimum value of p+q is

Correct answer

17

Step-by-step solution

aligned & & circum-radius distance of circum centre from any of the vertex & distance of a + b + c 4 from vertex D ( 0 ) [tetrahedron is regular] & Circumradius = 1 4 | a + b + c |= 1 4 a ^2+ b ^2+ c ^2+2( a b + b c + c a ) & = 1 4 k ^2+ k ^2+ k ^2+2 ( k ^2 2 + k ^2 2 + k ^2 2 ) = 1 4 6 k ^2 = 3 8 k & r R = 1 3 r = R 3 = k 24 & R = 3 8 k & r = k 24 aligned aligned & R^2+r^2= 5 12 k^2 & minimum value of p+q=17 aligned

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