First slide

Cartesian plane

Question

Locus of the centroid of the triangle whose vertices are $(a \cos t, a \sin t), (b \sin t, - b \cos t)$ and (1, 0); where t is $a$ parameter is

Moderate

A

$(3 x - 1)^{2} + (3 y)^{2} = a^{2} + b^{2}$
B

$(3 x + 1)^{2} + (3 y)^{2} = a^{2} + b^{2}$
C

$(3 x + 1)^{2} + (3 y)^{2} = a^{2} - b^{2}$
D

$(3 x - 1)^{2} + (3 y)^{2} = a^{2} - b^{2}$

Solution

If $(h, k)$ is the centroid, then
$h = \frac{a \cos t + b \sin t + 1}{3}$
$k = \frac{a \sin t - b \cos t + 0}{3}$
$\Rightarrow (3 h - 1)^{2} + (3 k)^{2} = (a \cos t + b \sin t)^{2} + (a \sin t - b \cos t)^{2}$
Locus of $\begin{matrix} = a^{2} + b^{2} \\ (h, k) is (3 x - 1)^{2} + (3 y)^{2} = a^{2} + b^{2} \end{matrix}$

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