Question:

<p><strong>Step 1: Find the slope of the tangent at (1, 0)</strong></p><p>Given: y = log₍ₑ₎ x</p><p>Differentiating: dy/dx = 1/x</p><p>At x = 1: dy/dx = 1/1 = 1</p><p>So the slope of the tangent is m₁ = 1</p><p><strong>Step 2: Find the slope of the normal</strong></p><p>The slope of the normal is the negative reciprocal of the tangent slope:</p><p>m₂ = -1/m₁ = -1/1 = -1</p><p><strong>Step 3: Write the equation of the normal</strong></p><p>Using point-slope form with point (1, 0) and slope m₂ = -1:</p><p>y - 0 = -1(x - 1)</p><p>y = -x + 1</p><p>Or: x + y = 1</p><p><strong>Step 4: Find the intercepts of the normal with the coordinate axes</strong></p><p>X-intercept (set y = 0): x + 0 = 1 ⟹ x = 1, giving point A(1, 0)</p><p>Y-intercept (set x = 0): 0 + y = 1 ⟹ y = 1, giving point B(0, 1)</p><p><strong>Step 5: Find the area of the triangle formed by the axes and the normal</strong></p><p>The triangle is formed by:</p><p>• Origin O(0, 0)</p><p>• Point A(1, 0) on the x-axis</p><p>• Point B(0, 1) on the y-axis</p><p>Area = (1/2) × base × height = (1/2) × 1 × 1 = 1/2 sq unit</p><p><strong>∴ Answer: C</strong></p>