<p>The locus of 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 :</p>
<p style="display:inline">(3x - 1)<sup>2</sup> + (3y)<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup></p>
<p style="display:inline">(3x + 1)<sup>2</sup> + (3y)<sup>2</sup> = a<sup>2</sup> - b<sup>2</sup></p>
<p style="display:inline">(3x - 1)<sup>2</sup> + (3y)<sup>2</sup> = a<sup>2</sup> - b<sup>2</sup></p>
<p style="display:inline">(3x + 1)<sup>2</sup> + (3y)<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup></p>
Step-by-Step Solution
Key Concept: The centroid of a triangle is found using (x,y) = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). After substituting the three vertices and expressing the parametric coordinates in terms of the centroid, eliminate the parameter t by squaring and adding the resulting equations to obtain the locus equation.
<p>Triangle vertices are<br />
A(x<sub>1</sub>, y<sub>1</sub>) = (a cos t, a sin t)<br />
B (x<sub>2</sub>, y<sub>2</sub>) = (b sin t, - b cos t)<br />
C (x<sub>3</sub>, y<sub>3</sub>) = (1, 0)<br />
Let the centroid point is (x, y)<br />
We know that,<br />
(x, y) = <span class="math-tex">$\left(\frac{{x}_{{1}}+{x}_{{2}}+{x}_{{3}}}{{3}}, \frac{{y}_{{1}}+{y}_{\mathbf{2}}+{y}_{3}}{{3}}\right)$</span><br />
(x, y) = <span class="math-tex">$\left(\frac{a \cos t+b \sin t+1}{3}, \frac{a \sin t-b \cos t+0}{3}\right)$</span><br />
3x = a cos t + b sin t + 1, 3y = a sin t - b cos t<br />
3x - 1 = a cos t + b sin t, 3y = a sin t - b cos t<br />
On squaring and adding both side we get,<br />
(3x - 1)<sup>2</sup> + (3y)<sup>2</sup> = (a cost + b sin t)<sup>2</sup> + (a sin t + b cos t)<sup>2</sup><br />
<span class="math-tex">$\Rightarrow$</span> (3x - 1)<sup>2</sup> + 9y<sup>2</sup> = a<sup>2</sup>cos<sup>2</sup>t + a<sup>2</sup>sin<sup>2</sup>t + 2an sin t cos t + a<sup>2</sup>sin<sup>2</sup> t + b<sup>2</sup>cos<sup>2</sup> t - 2 ab sin t cos t<br />
<span class="math-tex">$\Rightarrow$</span> (3x - 1)<sup>2</sup> + 9y<sup>2</sup> = a<sup>2</sup>cos<sup>2</sup>t + b<sup>2</sup>sin<sup>2</sup> t + a<sup>2</sup>sin<sup>2</sup> t + b<sup>2</sup>cos<sup>2</sup> t<br />
<span class="math-tex">$\Rightarrow$</span> (3x - 1)<sup>2</sup> + 9y<sup>2</sup> = a<sup>2</sup>cos<sup>2</sup> t + a<sup>2</sup>sin<sup>2</sup> t + b<sup>2</sup>sin<sup>2</sup> t + b<sup>2</sup> cos<sup>2</sup> t<br />
<span class="math-tex">$\Rightarrow$</span> (3x - 1)<sup>2</sup> + 9y<sup>2</sup> = a<sup>2</sup> (cos<sup>2</sup> t + sin<sup>2</sup> t) + b<sup>2 </sup>(sin<sup>2</sup> t + cos<sup>2</sup> t)<br />
<span class="math-tex">$\Rightarrow$</span> (3x - 1)<sup>2</sup> + (3y)<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup></p>
Correct Answer: A