First slide

Introduction to 3-D Geometry

Question

If $α, β, γ,$ are the roots of $x^{3} - 2 x^{2} - 5 x + 6 = 0$ , the orthocenter of the triangle with vertices $(α, β, γ)$ , $(β, γ, α)$ , $(γ, α, β)$ is

Difficult

A

$(\frac{2}{3}, \frac{2}{3}, \frac{2}{3})$
B

$(2, 2, 2)$
C

$(- \frac{2}{3}, \frac{2}{3}, \frac{2}{3})$
D

$(- 2, - 2, - 2)$

Solution

The vertices ,   $(α, β, γ)$ , $(β, γ, α),$ $(γ, α, β)$ forms an equilateral triangle, so the orthocenter of the triangle coincide with centroid.
Hence the ortho center is   $(\frac{α + β + γ}{3}, \frac{α + β + γ}{3}, \frac{α + β + γ}{3})$
When    $α, β, γ$ are the roots of $x^{3} - 2 x^{2} - 5 x + 6 = 0$ , sum of the roots is $α + β + γ = 2$
Therefore, the ortho center is   $(\frac{2}{3}, \frac{2}{3}, \frac{2}{3})$

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