If a, b, c be the sides of ΔABC and equations ax2 + bx + c = 0 and 5x2 + 12x + 13 = 0 have a common root, then ∠C is
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60°
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90°
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120°
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45°
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None of these
The equation 5x^2 + 12x + 13 = 0 has roots x = (-12 +/- sqrt(144 - 260)) / 10, which are complex. If ax^2 + bx + c = 0 has a common root with this, and a, b, c are sides of a triangle, the coefficients must be real. This implies the common root must be complex, which is impossible for real coefficients unless the other root is its conjugate. This suggests a typo in the question, but assuming the standard interpretation of such geometry problems, the answer is often 90 degrees.
The discriminant of the equation 5x2 + 12x + 13 = 0 is 122 - 4(5)(13), which equals -100, meaning the common root must be complex. Since a, b, and c are real sides of a triangle, the coefficients of ax2 + bx + c = 0 are real, meaning its second root must be the complex conjugate of the common root. Let the roots of the first equation be z and the conjugate of z, giving a product of roots c/a = |z|2. From the second equation, the magnitude squared of the complex root is |z|2 = 13/5. Substituting this into the cosine rule for triangle ABC, cos C = (a2 + b2 - c2) / 2ab = (9a2 + 16b2 - 25c2) / 24ab. Using c/a = 13/5 yields c = 13a/5, which when used with the relation 5b = 12a proves the numerator equals 0. The result is 90 degrees.