<p>The equation of the tangent to the circle \(x^2 + y^2 + 4x - 4y + 4 = 0\) which make equal intercepts on the positive coordinate axes, is</p>
Step-by-Step Solution
Key Concept: A line making equal intercepts on positive axes has form x + y = a (where a > 0). Substitute this into the tangent condition that distance from center equals radius to find a.
<p><strong>Step 1:</strong> Rewrite the circle equation in standard form.</p><p>x² + y² + 4x - 4y + 4 = 0</p><p>(x² + 4x + 4) + (y² - 4y + 4) - 4 = 0</p><p>(x + 2)² + (y - 2)² = 4</p><p>Center: C(-2, 2), Radius: r = 2</p><p><strong>Step 2:</strong> Write the equation of a line making equal intercepts on positive axes.</p><p>If intercepts are equal on positive axes: x/a + y/a = 1 (where a > 0)</p><p>This simplifies to: <strong>x + y = a</strong></p><p><strong>Step 3:</strong> Apply the tangency condition.</p><p>Distance from center C(-2, 2) to line x + y - a = 0 must equal radius 2.</p><p>Distance = |(-2) + 2 - a|/√(1² + 1²) = |-a|/√2 = |a|/√2</p><p><strong>Step 4:</strong> Set distance equal to radius.</p><p>|a|/√2 = 2</p><p>|a| = 2√2</p><p>Since a > 0: <strong>a = 2√2</strong></p><p><strong>Step 5:</strong> Write the tangent equation.</p><p><strong>x + y = 2√2</strong> or <strong>x + y - 2√2 = 0</strong></p><p>∴ Answer: B</p>
Correct Answer: B