Question:
<p>If a rectangular hyperbola <span class="math-tex">\((x-1)(y-2)=4\)</span> cuts a circle <span class="math-tex">\(x^2+y^2+2 g x+2 f y+c=0\)</span> at points <span class="math-tex">\((3,4),(5,3),(2,6)\)</span> and <span class="math-tex">\((-1,0)\)</span>, then the value of <span class="math-tex">\((g+f)\)</span> is equal to</p>
<p style="display:inline">-9</p>
<p style="display:inline">9</p>
<p style="display:inline">8</p>
<p style="display:inline">-8</p>
Step-by-Step Solution
Key Concept: The arithmetic mean of the coordinates of the four points where a circle intersects a rectangular hyperbola is equal to the midpoint of the line segment joining the centers of the two curves.
<p>We know that if a circle cuts a rectangular hyperbola then arithmetic mean of points of intersections is the mid-point of centre of hyperbola and circle.<br />
<span class="math-tex">$ \text { So, } \frac{3+5+2+(-1)}{4}=\frac{-g+1}{2}, \frac{4+3+6+0}{4}=\frac{-f+2}{2} $</span><br />
<span class="math-tex">$ \Rightarrow g+f=\left(-\frac{7}{2}\right)+\left(-\frac{9}{2}\right)=-8 $</span></p>
Correct Answer: D