Answer: e = sqrt(6)/2.
- A e = 3/2
- B e = sqrt(6)/2
- C e = sqrt(3)/2
- D e = 5/4
Correct answer: B. e = sqrt(6)/2
Explanation: Latus rectum = 2b<sup>2</sup>/a, transverse axis = 2a. Given 2b<sup>2</sup>/a = (1/2)(2a) = a, so 2b<sup>2</sup> = a<sup>2</sup>, meaning b<sup>2</sup> = a<sup>2</sup>/2. Then e<sup>2</sup> = 1 + b<sup>2</sup>/a<sup>2</sup> = 1 + 1/2 = 3/2, so e = sqrt(3/2) = sqrt(6)/2.
All four conics form a single family distinguished only by eccentricity: a circle is the most "closed" (e=0), an ellipse is an elongated closed curve, a parabola is the borderline open curve (e=1), and a hyperbola has two separate open branches (e>1).
Concept context
Study circles, parabolas, ellipses, and hyperbolas as curves formed by intersecting a plane with a double cone, with their standard equations and key properties.