Trigonometry & Inverse Trigonometry
Properties of Triangles
Grade 11
Question:
<p>Let <i>a</i>, <i>b</i>, <i>c</i> denotes side lengths of △<i>ABC</i>. If <i>a</i>, <i>b</i>, <i>c</i> are the roots of \(8x^3 + (\lambda + 2)x^2 - (2k + \lambda)x - 27 = 0\) such that \(\lambda^2 + 2\lambda(k+1) + 4k = 2^3 \cdot 3^5\), then which of the following is(are) <b>correct</b>?</p>
<p>Circumradius of triangle <i>ABC</i> is \(\dfrac{\sqrt{3}}{2}\).</p>
<p>Distance between orthocentre and side <i>AB</i> is \(\dfrac{\sqrt{3}}{4}\).</p>
<p>Distance between orthocentre and circumcentre of △<i>ABC</i> is \(\dfrac{\sqrt{3}}{4}\).</p>
<p>Distance between orthocentre and side <i>BC</i> is \(\dfrac{\sqrt{3}}{2}\).</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to relate the cubic's coefficients to triangle sides a, b, c, then apply the constraint equation to find specific values of λ and k, finally verify triangle inequality and angle conditions.
<p><strong>Step 1: Apply Vieta's formulas to the cubic 8x³ + (λ+2)x² - (2k+λ)x - 27 = 0</strong></p><p>Dividing by 8: x³ + [(λ+2)/8]x² - [(2k+λ)/8]x - 27/8 = 0</p><p>For roots a, b, c:</p><ul><li>a + b + c = -(λ+2)/8</li><li>ab + bc + ca = -(2k+λ)/8</li><li>abc = 27/8</li></ul><p><strong>Step 2: Use the constraint λ² + 2λ(k+1) + 4k = 2³·3⁵ = 8·243 = 1944</strong></p><p>Rearrange: λ² + 2λk + 2λ + 4k = 1944</p><p>λ² + 2λ(k+1) + 4k = 1944</p><p><strong>Step 3: For a valid triangle with abc = 27/8, try a = 3/2, b = 3/2, c = 3/2 (equilateral)</strong></p><p>This gives a + b + c = 9/2, so -(λ+2)/8 = 9/2 → λ + 2 = -36 → λ = -38</p><p>And ab + bc + ca = 27/4, so -(2k+λ)/8 = 27/4 → 2k + λ = -54</p><p>With λ = -38: 2k - 38 = -54 → k = -8</p><p><strong>Step 4: Verify constraint: (-38)² + 2(-38)(-8+1) + 4(-8) = 1444 + 2(-38)(-7) - 32 = 1444 + 532 - 32 = 1944 ✓</strong></p><p><strong>Step 5: For equilateral triangle a = b = c = 3/2:</strong></p><ul><li>A = B = C = 60°</li><li>All triangle inequalities satisfied</li><li>cos A = 1/2 (matches 60°)</li></ul><p>∴ Answer: A, B, C</p>
Correct Answer: A,B,C