Geometry

Question Analysis  

The main focus is to derive the equation of a parabola using its definition: the set of points equidistant from a fixed point (focus) and a fixed line (directrix).  

 

Key Concept Explanation  

A parabola is defined as the set of points where the distance to the focus equals the distance to the directrix. For a vertical parabola with focus and directrix , the standard form is .  

 

Step-by-Step Solution  

1. Identify focus and directrix:  

Focus: , Directrix: .  

2. Find the vertex :  

The vertex is the midpoint between the focus and directrix:  

.

3. Calculate :  

Distance from vertex to focus (or directrix): 
.

4. Write the equation:  

Substitute , ,

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