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  • Let two vertices of a triangle ABC be (2,4,6)  and ( 0,-2,-5) and its centroid be (2, 1, –1). If the image of the third vertex in the plane x + 2y + 4z = 11 is  <im

Let two vertices of a triangle ABC be (2,4,6)  and ( 0,-2,-5) and its centroid be (2, 1, –1). If the image of the
third vertex in the plane x + 2y + 4z = 11 is  (\alpha ,\beta ,\gamma )  then  \alpha \beta +\beta \gamma +\gamma \alpha  is equal to

 

Option: 1

76


Option: 2

74


Option: 3

70


Option: 4

72


Answers (1)

best_answer

Given Two vertices of Triangle
A(2,4,6) and B(0, -2, -5)  and if centroid G(2,1, -1) . Let Third vertices be (x, y,z)

Now  \frac{2+0+x}{3}=2,\frac{4-2+y}{3}=1,\frac{6-z+5}{3}=-1

x=4,\, y=1, \, z=-1

Third vertices C(4,1, -4) Now, Image of vertices C(4,1,–4) in the given plane is D=(\alpha ,\beta ,\gamma )

Now

\begin{aligned} & \frac{\alpha-4}{1}=\frac{\beta-1}{2}=\frac{\gamma+4}{4}=-2 \frac{(4+2-16-11)}{1+4+16} \\ & \frac{\alpha-4}{1}=\frac{\beta-1}{2}=\frac{\gamma+4}{4}=\frac{42}{21} \Rightarrow 2 \\ & \alpha=6, \beta=5, \gamma=4 \end{aligned}

Then \alpha \beta+\beta \gamma+\gamma \alpha

=(6\times 5)+(5\times 4)+(4 \times6)\\\,\: \, =30+20+24\\=\: 74

 

Posted by

Deependra Verma

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