Question

The eccentricity of the parabola is :

• A. 1
• B. 0
• C. Less than 1
• D. Greater than 1

Hint:

Remember the scale of eccentricity for conics:
- Circle: \(e = 0\)
- Ellipse: \(0 < e < 1\)
- Parabola: \(e = 1\)
- Hyperbola: \(e > 1\)

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to determine the value of the eccentricity of a parabola.
Eccentricity is a parameter associated with every conic section that measures how much it deviates from being circular.
Key Formula or Approach: By definition, a conic section is the locus of a point \(P\) whose distance from a fixed point (focus, \(F\)) and a fixed line (directrix, \(d\)) are in a constant ratio:
\[ e = \frac{PF}{PM} \] where \(PM\) is the perpendicular distance from \(P\) to the directrix.

Step 2: Detailed Explanation:

• Conic sections are classified according to the value of their eccentricity \(e\):
- Circle: \(e = 0\)
- Ellipse: \(0 < e < 1\) (eccentricity is less than 1)
- Parabola: \(e = 1\) (eccentricity is equal to 1)
- Hyperbola: \(e > 1\) (eccentricity is greater than 1)

• For a parabola, the definition states that any point on it is equidistant from the focus and the directrix.

• Since \(PF = PM\), the ratio is:
\[ e = \frac{PF}{PM} = 1 \]

• Therefore, the eccentricity of any parabola is always exactly 1.

Step 3: Final Answer:
The eccentricity of the parabola is 1.