Geometry

Question Analysis  

The main focus is to derive the equation of a parabola using the definition that all points on the parabola are equidistant from the focus and the directrix. This requires identifying the vertex, orientation, and parameter .  

 

Key Concept Explanation  

A parabola is the set of points where the distance to the focus equals the distance to the directrix. For a vertical parabola (focus and directrix aligned vertically), the vertex is the midpoint between the focus and directrix, and the standard form is , where is the distance from the vertex to the focus.  

 

Step-by-Step Solution  

1. Find the vertex :  

The vertex is the midpoint between the focus and the directrix (a horizontal line).  

-coordinate of vertex: .  

-coordinate of vertex: (same as focus, since aligned vertically).  

Vertex: .  

 

2. Calculate :  

is the distance from the vertex to the focus (negative since the focus is below the vertex for a downward-opening parabola).  

(or , with negative sign for downward direction).  

 

3. Write the equation:  

Substitute , , and into the vertical parabola form

Click "Show Answer" to reveal the answer and analysis