Coplanarity (2)

Geometry Level 4

Find the sum of all possible values of n n such that points E ( − n , 1 , 0 ) E(-n,1,0) , M ( 0 , − n , 1 ) M(0,-n,1) , T ( 4 , − 17 , 8 − n ) T(4,-17,8-n) and the origin are coplanar.


The answer is 8.

Tommy Li by Tommy Li , 22, Hong Kong

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1 solution

Miles Koumouris
Dec 10, 2017

A plane has the equation z = a x + b y + c , z=ax+by+c, and we are given that 0 = a ( − n ) + b ( 1 ) + c 1 = a ( 0 ) + b ( − n ) + c 8 − n = a ( 4 ) + b ( − 17 ) + c 0 = a ( 0 ) + b ( 0 ) + c . \begin{aligned} 0&=a(-n)+b(1)+c\\ 1&=a(0)+b(-n)+c\\ 8-n&=a(4)+b(-17)+c\\ 0&=a(0)+b(0)+c. \end{aligned} Solving these equations gives n = 4 , − 3 + 2 , 3 + 2 , n=4,\;\; -\sqrt{3}+2,\;\; \sqrt{3}+2, and therefore, the answer is 8 \boxed{8} .

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