Question:
<p>The circle x<sup>2</sup> + y<sup>2</sup> - 4x - 4y + 4 = 0 is inscribed in a triangle that has two of its sides along the coordinate axes. If the locus of the circumcentre of the triangle is x + y - xy + k<span class="math-tex">\(\sqrt{x^{2}+y^{2}}\)</span> = 0, then k =</p>
<p style="display:inline">3</p>
<p style="display:inline">1</p>
<p style="display:inline">2</p>
<p style="display:inline"><span class="math-tex">\(\sqrt{2}\)</span></p>
Step-by-Step Solution
Key Concept: Utilize the relationship between a triangle's area, its inradius, and its semi-perimeter ($A = rs$) along with the fact that the circumcentre of a right-angled triangle is the midpoint of its hypotenuse.
<html><body><p>x<sup>2</sup> + y<sup>2</sup> - 4x - 4y + 4 = 0<br/>
<span class="math-tex">$\Leftrightarrow$</span> (x - 2)<sup>2</sup> + (y - 2)<sup>2</sup> = 4<br/>
<span class="math-tex">$\Rightarrow$</span> centre = (2, 2), radius = 2<br/>
<img alt="" data-imgur-src="Ds7CbvF.png" height="159" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1623761652-q8sbg7.jpg" width="179"/><br/>
Let OAB be a variable triangle circumscribing the circle.<br/>
Let A = (a, 0), B = (0, b)<br/>
area of <span class="math-tex">$\triangle$</span>OAB = area of <span class="math-tex">$\triangle$</span>OSA + area of <span class="math-tex">$\triangle$</span>ASB + area of <span class="math-tex">$\triangle$</span>BSO<br/>
<span class="math-tex">$\Rightarrow \frac{1}{2} a b=\frac{1}{2}\left[(a)(2)+2 \sqrt{a^{2}+b^{2}}+(2)(b)\right]$</span><br/>
<span class="math-tex">$\Rightarrow a b=2\left(a+b+\sqrt{a^{2}+b^{2}}\right)$</span> ...(i)<br/>
Let M = (p, q) be the circumcentre.<br/>
<span class="math-tex">$\Rightarrow p=\frac{a}{2}, q=\frac{b}{2} \Rightarrow$</span> a = 2p, b = 2q<br/>
Substituting the values of a and b in (i), we get<br/>
4pq = 2[2(p + q) + 2<span class="math-tex">$\sqrt{p^{2}+q^{2}}$</span>]<br/>
<span class="math-tex">$\Rightarrow$</span> (p, q) satisfies xy = x + y + <span class="math-tex">$\sqrt{x^{2}+y^{2}}$</span><br/>
<span class="math-tex">$\Rightarrow$</span> x + y - xy + <span class="math-tex">$\sqrt{x^{2}+y^{2}}$</span> = 0 <span class="math-tex">$\Rightarrow$</span> k = 1</p></body></html>
Correct Answer: B