Question:
<p>Let A (-3, 2) and B (-2, 1) be the vertices of a triangle ABC. If the centroid of triangle ABC lies on the line 3x + 4y + 2 = 0, then the locus of vertex C is :</p>
<p style="display:inline">3x + 4y + 5=0</p>
<p style="display:inline">3x + 4y + 3 = 0</p>
<p style="display:inline">4x + 3y + 5 = 0</p>
<p style="display:inline">4x + 3y + 3 = 0</p>
Step-by-Step Solution
Key Concept: Express the centroid in terms of the unknown vertex coordinates using the formula ((x1+x2+x3)/3, (y1+y2+y3)/3) and substitute these expressions into the given line equation to find the locus.
<html><body><p>Let C = (x<sub>1</sub>, y<sub>1</sub>)<br/>
Centroid, E = <span class="math-tex">$\left(\frac{x_{1}-5}{3}, \frac{y_{1}+3}{3}\right)$</span><br/>
Since centroid lies on the line<br/>
3x + 4y + 2 = 0<br/>
<span class="math-tex">$\therefore$</span> 3 <span class="math-tex">$\left(\frac{x_{1}-5}{3}\right)$</span> + 4 <span class="math-tex">$\left(\frac{y_{1}+3}{3}\right)$</span> + 2 = 0<br/>
<span class="math-tex">$\Rightarrow$</span> 3x<sub>1</sub> + 4y<sub>1</sub> + 3 = 0<br/>
Hence vertex (x<sub>1</sub>, y<sub>1</sub>) lies on the line<br/>
3x + 4y + 3 = 0<br/>
<img alt="" data-imgur-src="qjESU30.png" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1619691795-5uqc6s.jpg" style="width: 200px; height: 129px;"/></p></body></html>
Correct Answer: B