Question

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 write the value of tan θ.

Answer 1

The centroid of a triangle with vertices \[\left( x_1 , y_1 \right), \left( x_2 , y_2 \right)\text { and } \left( x_3 , y_3 \right)\] is given below: \[\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)\].

Therefore, the centre of the triangle having vertices (0, 0), (cos θ, sin θ) and (sin θ, − cos θ) is

\[\left( \frac{0 + cos\theta + sin\theta}{3}, \frac{0 + sin\theta - cos\theta}{3} \right) \equiv \left( \frac{cos\theta + sin\theta}{3}, \frac{sin\theta - cos\theta}{3} \right)\]

This point lies on the line y = 2x. 

\[\frac{sin\theta - cos\theta}{3} = 2 \times \frac{cos\theta + sin\theta}{3}\]

\[ \Rightarrow sin\theta - cos\theta = 2cos\theta + 2sin\theta\]

\[ \Rightarrow tan\theta = - 3\]

∴ tanθ = −3