Circles
Touching circles
Grade 11
Question:
<p>Let <em>C</em> be the circle with centre at (1, 1) and radius = 1. If <em>T</em> is the circle centred at (0, <em>y</em>), passing through origin and touching the circle <em>C</em> externally, then the radius of <em>T</em> is equal to</p>
<p>\(\dfrac{1}{2}\)</p>
<p>\(\dfrac{1}{4}\)</p>
<p>\(\dfrac{\sqrt{3}}{\sqrt{2}}\)</p>
<p>\(\dfrac{\sqrt{3}}{2}\)</p>
Step-by-Step Solution
Key Concept: For two circles touching externally, the distance between their centers equals the sum of their radii. Use this condition along with the constraint that circle T passes through the origin to find its radius.
<p><strong>Step 1:</strong> Identify circle C: center (1, 1), radius = 1</p><p><strong>Step 2:</strong> Identify circle T: center (0, y), passes through origin (0, 0). Since T passes through origin, its radius r = |y|</p><p><strong>Step 3:</strong> For external tangency, distance between centers = sum of radii:</p><p>$$\sqrt{(1-0)^2 + (1-y)^2} = 1 + |y|$$</p><p><strong>Step 4:</strong> Square both sides:</p><p>$$1 + (1-y)^2 = (1 + |y|)^2$$</p><p><strong>Step 5:</strong> For y > 0: </p><p>$$1 + 1 - 2y + y^2 = 1 + 2y + y^2$$</p><p>$$2 - 2y = 1 + 2y$$</p><p>$$1 = 4y$$</p><p>$$y = \frac{1}{4}$$</p><p><strong>Step 6:</strong> Therefore, radius of T = |y| = <strong>1/4</strong></p><p>∴ Answer: B</p>
Correct Answer: B