Question:
<p>Let <span class="math-tex">\(A B C D\)</span> and <span class="math-tex">\(A E F G\)</span> be squares of side 4 and 2 units, respectively. The point <span class="math-tex">\(E\)</span> is on the line segment <span class="math-tex">\(A B\)</span> and the point <span class="math-tex">\(F\)</span> is on the diagonal <span class="math-tex">\(A C\)</span>. Then the radius <span class="math-tex">\(r\)</span> of the circle passing through the point <span class="math-tex">\(F\)</span> and touching the line segments <span class="math-tex">\(B C\)</span> and <span class="math-tex">\(C D\)</span> satisfies:</p>
<p style="display:inline"><span class="math-tex">\(2 r^{2}-8 r+7=0\)</span></p>
<p style="display:inline"><span class="math-tex">\(r^{2}-8 r+8=0\)</span></p>
<p style="display:inline"><span class="math-tex">\(2 r^{2}-4 r+1=0\)</span></p>
<p style="display:inline"><span class="math-tex">\(r=1\)</span></p>
Step-by-Step Solution
Key Concept: Represent the circle's center as (4-r, 4-r) based on its tangency to the lines BC and CD, then equate its distance from point F(2, 2) to the radius r.
<p><img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1775193367-486c8b.jpg" style="height:175px; width:200px" /><br />
<img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1775193396-jkpwh7.jpg" style="height:150px; width:200px" /><br />
<span class="math-tex">\(O F^{2}=r^{2}\)</span><br />
<span class="math-tex">\((2-r)^{2}+(2-r)^{2}=r^{2}\)</span><br />
<span class="math-tex">\(r^{2}-8 r+8=0\)</span></p>
Correct Answer: B