First slide

Equation of line in Straight lines

Question

If $A (\cos α, \sin α), B (\sin α, - \cos α), C (1, 2)$ are the varties $△ A B C$ then as a varies the locus of its centroid, is

Moderate

A

$x^{2} + y^{2} - 2 x - 4 y + 1 = 0$
B

$3 (x^{2} + y^{2}) - 2 x - 4 y + 1 = 0$
C

$x^{2} + y^{2} - 2 x - 4 y + 3 = 0$
D

none of these

Solution

let $(h, k)$ be the centroid of the triangle having
$A (\cos α, \sin α), B (\sin α, - \cos α) and C (1, 2)$ Then $h = \frac{\cos α + \sin α + 1}{3} and k = \frac{\sin α - \cos α + 2}{3}$
$\Rightarrow 3 h - 1 = \cos α + \sin α$
$\begin{aligned} \Rightarrow 9 (h^{2} + k^{2}) - 6 h - 12 k + 3 = 0 \\ \Rightarrow 3 (h^{2} + k^{2}) - 2 h - 4 k + 1 = 0 \end{aligned}$
Hence, the locus of $(h, k) is 3 (x^{2} + y^{2}) - 2 x - 4 y + 1 = 0$

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