<p><b>Paragraph for Question nos. 620 and 621</b><br>Equation of an altitude of an equilateral triangle is \(\sqrt{3}x + y = 2\sqrt{3}\) and one of its vertex is \((3, \sqrt{3})\). Then:</p><p>If orthocentre \(H(a, b)\) of the triangle lies in the first quadrant, then \(a^2 + b^2\) is equal to:</p>
Step-by-Step Solution
Key Concept: In an equilateral triangle, the orthocenter coincides with the centroid. Since an altitude is given and one vertex is known, find the orthocenter by using the property that it lies on the altitude and is equidistant from all vertices in a specific geometric configuration.
<p><strong>Step 1:</strong> Verify the given vertex A(3, √3) doesn't lie on altitude √3x + y = 2√3. Check: √3(3) + √3 = 4√3 ≠ 2√3. ✓ The vertex is not on this altitude.</p><p><strong>Step 2:</strong> In an equilateral triangle, all three altitudes meet at the orthocenter H, and by symmetry, H is equidistant from all sides. The orthocenter lies on the given altitude √3x + y = 2√3.</p><p><strong>Step 3:</strong> The altitude from vertex A(3, √3) must be perpendicular to the opposite side. The given altitude has slope -√3, so the perpendicular altitude through A has slope 1/√3. Equation: y - √3 = (1/√3)(x - 3), which gives x - √3y = 0 or x = √3y.</p><p><strong>Step 4:</strong> Find orthocenter H by solving the system:<br>√3x + y = 2√3 ... (1)<br>x = √3y ... (2)<br>Substitute (2) into (1): √3(√3y) + y = 2√3 → 3y + y = 2√3 → 4y = 2√3 → y = √3/2<br>Then x = √3 · (√3/2) = 3/2</p><p><strong>Step 5:</strong> Orthocenter H(3/2, √3/2) lies in the first quadrant. Calculate:<br>a² + b² = (3/2)² + (√3/2)² = 9/4 + 3/4 = 12/4 = 3</p><p>∴ Answer: <strong>D (a² + b² = 3)</strong></p>
Correct Answer: D