Question

If \( \alpha, \beta \) are the two real roots of the 4th roots of unity, and \( \gamma, \delta \) are the other two roots of it, then the sum of the eccentricities of the conics \( |z - \alpha| + |z - \beta| = 4 \) and \( |z - \gamma| + |z - \delta| = 6 \) is:

• A. \( \frac{5}{6} \)
• B. \( \frac{5}{12} \)
• C. \( \frac{3}{7} \)
• D. \( \frac{4}{5} \)

Hint:

To solve problems with conics, focus on the properties of the roots of unity and eccentricity formulas for conic sections.

Answer 1

The Correct Option is A

We are given the sum of the eccentricities for two conics and need to calculate the result. The formula for the eccentricities involves the distance between the roots, and we apply this to get the desired value. By solving the conditions, we find the correct eccentricity sum.