Question:
<p>Four distinct points <span class="math-tex">\((2 k, 3 k),(1,0),(0,1)\)</span> and <span class="math-tex">\((0,0)\)</span> lie on a circle for <span class="math-tex">\(k\)</span> equal to:</p>
<p style="display:inline"><span class="math-tex">\(\frac{2}{13}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{13}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{3}{13}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{5}{13}\)</span></p>
Step-by-Step Solution
Key Concept: Identify the equation of the circle passing through the origin and the axes intercepts (1,0) and (0,1), then substitute the fourth point into this equation.
<p>Given four points lie on circle.<br />
So, the equation of circle is<br />
<span class="math-tex">$(x-1)(x-0)+(y-0)(y-1)=0$</span><br />
<span class="math-tex">$\Rightarrow x^{2}-x+y^{2}-y=0$</span> ...(i)<br />
<span class="math-tex">$\because(2 k, 3 k)$</span> lie on (i), we get<br />
<span class="math-tex">$\Rightarrow(2 k)^{2}-2 k+(3 k)^{2}-3 k=0$</span><br />
<span class="math-tex">$\Rightarrow 13 k^{2}-5 k=0 $</span> <span class="math-tex">$\Rightarrow k(13 k-5)=0$</span><br />
<span class="math-tex">$\Rightarrow k=0$</span> or <span class="math-tex">$k=\frac{5}{13}$</span><br />
<span class="math-tex">$\therefore$</span> Value of <span class="math-tex">$k=\frac{5}{13}$</span></p>
Correct Answer: D