<p>There are two real tangents from the point (2, 1) to the circle \(x^2 + y^2 = 8\).</p><p><em>State whether the statement is true or false.</em></p>
Step-by-Step Solution
Key Concept: A point lies outside a circle if its distance from center exceeds the radius. Two real tangents exist only when the point is outside the circle.
<p><strong>Step 1:</strong> Identify the circle parameters. Circle equation: x² + y² = 8, so center C = (0, 0) and radius r = √8 = 2√2.</p><p><strong>Step 2:</strong> Calculate the distance from point P(2, 1) to center C(0, 0).</p><p>d = √[(2-0)² + (1-0)²] = √(4 + 1) = √5</p><p><strong>Step 3:</strong> Compare distance with radius.</p><p>√5 ≈ 2.236 and 2√2 ≈ 2.828</p><p>Since √5 < 2√2, we have d < r, meaning the point (2, 1) lies <strong>inside</strong> the circle.</p><p><strong>Step 4:</strong> Apply tangent condition. Two real tangents from an external point exist only when d > r. Since d < r here, <strong>no real tangents</strong> can be drawn.</p><p>∴ Answer: <strong>FALSE</strong> (The statement is false. No real tangents exist from (2, 1) to the circle.)</p>
Correct Answer: B