Question:
<p>Let (<span class="math-tex">\(\alpha\)</span>, <span class="math-tex">\(\beta\)</span>) be the centroid of the triangle formed by the lines 15x - y = 82, 6x - 5y = -4 and 9x + 4y = 17. Then <span class="math-tex">\(\alpha\)</span> + 2<span class="math-tex">\(\beta\)</span> and 2<span class="math-tex">\(\alpha\)</span> - <span class="math-tex">\(\beta\)</span> are the roots of the equation</p>
<p style="display:inline">x<sup>2</sup> - 10x + 25 = 0</p>
<p style="display:inline">x<sup>2</sup> - 14x + 48 = 0</p>
<p style="display:inline">x<sup>2</sup> - 13x + 42 = 0</p>
<p style="display:inline">x<sup>2 </sup>- 7x + 12 = 0</p>
Step-by-Step Solution
Key Concept: Determine the triangle's vertices by solving the three systems of linear equations, find the centroid using the coordinate averages, and then construct the quadratic equation using the sum and product of the transformed roots.
<p>After solving the equations, we get coordinates as (6, 8), (1, 2) and (5, -7)<br />
So centroid: (<span class="math-tex">\(\alpha\)</span>, <span class="math-tex">\(\beta\)</span>) is<br />
<span class="math-tex">\(\alpha\)</span> = <span class="math-tex">\(\frac{6+1+5}{3}\)</span> = 4 and <span class="math-tex">\(\beta\)</span> = <span class="math-tex">\(\frac{8+2-7}{3}\)</span> = 1<br />
<span class="math-tex">\(\alpha\)</span> + 2<span class="math-tex">\(\beta\)</span> = 6 and 2<span class="math-tex">\(\alpha\)</span> - <span class="math-tex">\(\beta\)</span> = 7<br />
So, required equation = x<sup>2</sup> - 13x + 42 = 0</p>
Correct Answer: C