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 $\left(x_1,y_1\right),\left(x_2,y_2\right)\mathrm{and}\left(x_3,y_3\right)$ 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
$\left(\frac{0+\mathrm{cos\theta }+\mathrm{sin\theta }}{3},\frac{0+\mathrm{sin\theta }-\mathrm{cos\theta }}{3}\right)\equiv \left(\frac{\mathrm{cos\theta }+\mathrm{sin\theta }}{3},\frac{\mathrm{sin\theta }-\mathrm{cos\theta }}{3}\right)$

This point lies on the line y = 2x.

$$\begin{gathered} \frac{\mathrm{sin\theta }-\mathrm{cos\theta }}{3}=2\times \frac{\mathrm{cos\theta }+\mathrm{sin\theta }}{3} \\ \Rightarrow \mathrm{sin\theta }-\mathrm{cos\theta }=2\mathrm{cos\theta }+2\mathrm{sin\theta } \\ \Rightarrow \mathrm{tan\theta }=-3 \end{gathered}$$

∴ tanθ = −3