Question:
<p>The equation of the circle passing through (1, 2) and the points of intersection of the circles x<sup>2</sup> + y<sup>2</sup> - 8x - 6y + 21 = 0 and x<sup>2</sup> + y<sup>2</sup> - 2x - 15 = 0 is</p>
<p style="display:inline">x<sup>2</sup> + y<sup>2</sup> - 6x - 4y + 9 = 0</p>
<p style="display:inline">x<sup>2</sup> + y<sup>2</sup> - 6x + 4y + 9 = 0</p>
<p style="display:inline">x<sup>2</sup> + y<sup>2</sup> + 6x - 2y + 9 = 0</p>
<p style="display:inline">x<sup>2</sup> + y<sup>2</sup> - 6x - 2y + 9 = 0</p>
Step-by-Step Solution
Key Concept: The equation of a circle passing through the intersection of two circles $S_1=0$ and $S_2=0$ is found using the family of circles equation $S_1 + kS_2 = 0$.
<p>Equation of the required circle passing through the point of intersection of the given circles is<br />
x<sup>2</sup> + y<sup>2</sup> - 8x - 6y + 21 + k(x<sup>2</sup> + y<sup>2 </sup>- 2x - 15) = 0<br />
Also, it passes through (1, 2).<br />
<span class="math-tex">\(\Rightarrow k=\frac{1}{2}\)</span><br />
Equation of required circle is<br />
x<sup>2</sup> + y<sup>2</sup> - 8x - 6y + 21 + <span class="math-tex">\(\frac{1}{2}\)</span>(x<sup>2</sup> + y<sup>2</sup> - 2x - 15) = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> x<sup>2</sup> + y<sup>2</sup> - 6x - 4y + 9 = 0</p>
Correct Answer: A