The sum of the eccentricities of two different conics is $\frac{3}{4}$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2016 - Paper 1

To solve this problem, we need to understand the eccentricities of different conics:

1. Circle: The eccentricity $e = 0$.
2. Ellipse: The eccentricity $0 < e < 1$.
3. Parabola: The eccentricity $e = 1$.
4. Hyperbola: The eccentricity $e > 1$.

Given that the sum of the eccentricities is $\frac{3}{4}$, we look for pairs of conics:

• Circle and Ellipse: Sum = $0 + e < 1$.
• Ellipse and Parabola: Sum = $e_1 + 1$, which cannot equal $\frac{3}{4}$ since $e_1 < 1$.
• Parabola and Hyperbola: Sum = $1 + e_2 > 1$.
• Hyperbola and Circle: Sum = $e_1 + 0 > 1$.

Since the only pair whose sum fits the condition of being less than 1 yet positive and less than 1 for the ellipse is the Circle and Ellipse. Thus, the answer is (A) Circle and ellipse.