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A line passes through the focus of the parabola y<sup>2</sup> = 4ax and is perpendicular to its axis. The chord it cuts on the parabola is called the latus rectum. If a = 4, find the endpoints of the latus rectum.

Answer: (4, 8) and (4, -8).

  • A (4, 8) and (4, -8)
  • B (4, 4) and (4, -4)
  • C (8, 4) and (8, -4)
  • D (2, 8) and (2, -8)

Correct answer: A. (4, 8) and (4, -8)

Explanation: With a=4, parabola is y²=16x; focus at (a,0)=(4,0). At x=4: y²=16(4)=64, y=±8. Latus rectum endpoints are (4,8) and (4,−8), and its length=2(2a)=16. Ans: (4,8) and (4,−8).

The Four Conics by EccentricityCircle (e=0)Ellipse (0<e<1)Parabola (e=1)Hyperbola (e>1)

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.

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