If the centroid of a triangle formed by the points 0-0-cos- sin- and sin-cos- lies on the line y-2 x-then wirte the value of tan-

Centroid of triangle formed by (0,0),(cos θ,sin θ) and (sin θ, cos θ) is =(x_1+x_2+x_3/3,y_1+y_2+y_3/3 ) =(cosθ+sinθ/3,sinθ cosθ/3 ) It lies on line y=2x,so si ...

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Standard XII

Mathematics

Centroid

If the centro...

Question

If the centroid of a triangle formed by the points (0,0), $(c o s θ, s i n θ)$ and $(s i n θ, - c o s θ)$ lies on the line y=2x,then wirte the value of $t a n θ$ .

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Solution

Centroid of triangle formed by

$(0, 0), (c o s θ, s i n θ)$ and $(s i n θ, - c o s θ)$ is

$= (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$

$= (\frac{c o s θ + s i n θ}{3}, \frac{s i n θ - c o s θ}{3})$

It lies on line y=2x,so

$\frac{s i n θ - c o s θ}{3} = 2 \frac{c o s θ + s i n θ}{3}$

$s i n θ - c o s θ = 2 c o s θ + 2 s i n θ - s i n θ - s i n θ 3 c o s θ t a n θ = - 3$

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If the centroid of a triangle formed by the points (0,0), (cos θ,sin θ) and (sin θ,−cos θ) lies on the line y=2x,then wirte the value of tan θ.

Centroid of triangle formed by (0,0),(cos θ,sin θ) and (sin θ,−cos θ) is =(x1+x2+x33,y1+y2+y33) =(cosθ+sinθ3,sinθ−cosθ3) It lies on line y=2x,so sinθ−cosθ3=2cosθ+sinθ3 sinθ−cosθ=2cosθ+2sinθ−sinθ−sinθ3 cosθ tanθ=−3

If the centroid of a triangle formed by the points (0,0), $(c o s θ, s i n θ)$ and $(s i n θ, - c o s θ)$ lies on the line y=2x,then wirte the value of $t a n θ$ .