Question

The vertices of a triangle are $A(x_1,x_1\tan {\theta }_1)$, $B(x_2,x_2\tan {\theta }_2)$ & $C(x_3,x_3\tan {\theta }_3)$ If the circumcentre of the triangle ABC is at the origin & $H(\overline{x},\overline{y})$ be its orthocentre, then $\frac{\overline{x}}{\overline{y}}=\frac{\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3}{\sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3}$

• A. True
• B. False

Answer 1

The correct option is B False

Orthocenter, Centroid and circumcenter are collinear with centroid lying in between and trisecting the other two

$3\overrightarrow{C}-2\overrightarrow{C}=\overrightarrow{0}$

$x$ coordinate of $0\to 3\left(\frac{x_1+x_2+x_3}{3}\right)=x_1+x_2+x_3$

$y$ coordinate of $0\to 3\left(\frac{x_1\tan {\theta }_1+x_2\tan {\theta }_2+x_3\tan {\theta }_3}{3}\right)$

$=x_1\tan {\theta }_1+x_2\tan {\theta }_2+x_3\tan {\theta }_3$

$\therefore \frac{x}{y}=\frac{x_1+x_2+x_3}{x_1\tan {\theta }_1+x_2\tan {\theta }_2+x_3\tan {\theta }_3}$

Hence both expression are not equal