Question:
<p>Let <span class="math-tex">\(C\)</span> be a circle with radius <span class="math-tex">\(\sqrt{10}\)</span> units and centre at the origin. Let the line <span class="math-tex">\(x+y=2\)</span> intersects the circle <span class="math-tex">\(C\)</span> at the points <span class="math-tex">\(P\)</span> and <span class="math-tex">\(Q\)</span>. Let <span class="math-tex">\(M N\)</span> be a chord of <span class="math-tex">\(C\)</span> of length 2 unit and slope -1. Then, a distance (in units) between the chord <span class="math-tex">\(P Q\)</span> and the chord <span class="math-tex">\(M N\)</span> is</p>
<p style="display:inline"><span class="math-tex">\(3-\sqrt{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\sqrt{2}-1\)</span></p>
<p style="display:inline"><span class="math-tex">\(2-\sqrt{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\sqrt{2}+1\)</span></p>
Step-by-Step Solution
Key Concept: The distance between parallel chords is found by calculating their individual perpendicular distances from the center and considering both their same-side and opposite-side configurations.
<p><img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1775193172-4nnm66.jpg" style="height:176px; width:180px" /><br />
Equation of the circle is<br />
<span class="math-tex">\(x^{2}+y^{2}=10\)</span><br />
<span class="math-tex">\(\therefore A N=\frac{M N}{2}=1\)</span><br />
<span class="math-tex">\(\therefore \ln \triangle {OAN}\)</span><br />
<span class="math-tex">\(\Rightarrow(O N)^{2}=(O A)^{2}+(A N)^{2}\)</span><br />
<span class="math-tex">\(\Rightarrow 10=(O A)^{2}+1 \Rightarrow O A=3\)</span><br />
Perpendicular distance of centre from<br />
<span class="math-tex">\(P Q=\frac{|0+0-2|}{\sqrt{2}}=\sqrt{2}\)</span><br />
Perpendicular distance between <span class="math-tex">\(M N\)</span> and<br />
<span class="math-tex">\(P Q=A O+\sqrt{2}\)</span> or <span class="math-tex">\(|O A-\sqrt{2}|\)</span><br />
<span class="math-tex">\(=3+\sqrt{2}\)</span> or <span class="math-tex">\(3-\sqrt{2}\)</span></p>
Correct Answer: A