AP EAMCET · Maths · Pair of Lines
The equation \(x^2-2 \sqrt{3} x+2=0\) represents two sides of a triangle. If the angle between them is \(\frac{\pi}{3}\), then the perimeter of that triangle is
- A \(2 \sqrt{3}+6\)
- B \(2 \sqrt{3}+\sqrt{6}\)
- C \(3 \sqrt{2}+6\)
- D \(3 \sqrt{2}+\sqrt{6}\)
Answer & Solution
Correct Answer
(B) \(2 \sqrt{3}+\sqrt{6}\)
Step-by-step Solution
Detailed explanation
Let the two sides be \(a\) and \(b\). \(a+b = 2\sqrt{3}\), \(ab = 2\) \(a^2+b^2 = (a+b)^2 - 2ab = (2\sqrt{3})^2 - 2(2) = 12 - 4 = 8\) Let the third side be \(c\). By Law of Cosines: \(c^2 = a^2+b^2 - 2ab \cos(\frac{\pi}{3}) = 8 - 2(2)(\frac{1}{2}) = 8 - 2 = 6\) \(c = \sqrt{6}\)…
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