Question

If $A(\cos \alpha ,\sin \alpha ,0),B(\cos \beta ,\sin \beta ,0)$, $C(\cos \gamma ,\sin \gamma ,0)$ are vertices of $\Delta ABC$ and let

$\cos \alpha +\cos \beta +\cos \gamma =3a$, $\sin \alpha +\sin \beta +\sin \gamma =3b$, then correct matching of the following is:

List : I	List : II
A. Circumcentre	$1.(3a,3b,0)$
B. Centroid	$2.(0,0,0)$
C. Ortho centre	$3.(a,b,0)$

• A. $432$
• B. $231$
• C. $123$
• D. $234$

Answer 1

Answer: (B)

$231$

We know, ${\sin }^2\alpha +{\cos }^2\alpha =1$
Similarly it is true for other three coordinates.
These are points on a circle of unit radius.
By observation $(0,0,0)$ would be the circumcentre of the circle.
Centroid $=\frac{(\cos {\alpha }_1+\cos {\alpha }_2+\cos {\alpha }_3)}{3},\frac{(\sin {\alpha }_1+\sin {\alpha }_2+\sin {\alpha }_3)}{3},0$
Centroid $=(\frac{3a}{3},\frac{3b}{3},0)$=$(a,b,0)$
Centroid divides line segment joining orthocentre and circumcentre in the ratio 2:1.
Orthocentre $=(3a,3b,0)$