<p>Let the tangents drawn to the circle, \(x^2 + y^2 = 16\) from the point \(P(0, h)\) meet the \(x\)-axis at points \(A\) and \(B\). If the area of \(\triangle APB\) is minimum, then \(h\) is equal to</p>
Step-by-Step Solution
Key Concept: The area of triangle APB depends on the distance h from P to the x-axis and the distance between tangent points A and B on the x-axis. Using the tangent geometry and optimizing the area function, the minimum occurs at a specific value of h that can be found using calculus or geometric relationships.
<p><strong>Step 1:</strong> From point P(0, h) on the y-axis, draw tangents to circle x² + y² = 16 (radius r = 4).</p><p><strong>Step 2:</strong> The length of tangent from P(0, h) to the circle is: PT = √(h² - 16)</p><p><strong>Step 3:</strong> Let T be a point of tangency. Then OT ⊥ PT, where O is the origin. In right triangle OPT: tan(∠OPT) = 4/√(h² - 16)</p><p><strong>Step 4:</strong> The tangent meets the x-axis at A and B. By symmetry about the y-axis, if tangent touches at T, it meets x-axis at distance x = 4h/√(h² - 16) from origin.</p><p><strong>Step 5:</strong> Therefore, AB = 2 · (4h/√(h² - 16)) = 8h/√(h² - 16)</p><p><strong>Step 6:</strong> Area of △APB = (1/2) · AB · h = (1/2) · (8h/√(h² - 16)) · h = 4h²/√(h² - 16)</p><p><strong>Step 7:</strong> Let A = 4h²/√(h² - 16). To minimize, take dA/dh = 0:</p><p>dA/dh = [8h√(h² - 16) - 4h² · h/√(h² - 16)]/(h² - 16) = 0</p><p><strong>Step 8:</strong> 8h(h² - 16) - 4h³ = 0 → 8h³ - 128h - 4h³ = 0 → 4h³ = 128h → h² = 32 → h = 4√2</p><p>∴ Answer: A</p>
Correct Answer: A