Geometry

Question Analysis  

The main focus is to derive the equation of a horizontal parabola using the definition that all points on the parabola are equidistant from the focus and the directrix. This requires finding 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 horizontal parabola (focus and directrix aligned horizontally), 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 vertical line).  

-coordinate of vertex: .  

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

Vertex: .  

 

2. Calculate :  

is the distance from the vertex to the focus (positive since the focus is to the right of the vertex for a right-opening parabola).  

.  

 

3. Write the equation:  

Substitute , , and into the horizontal parabola form

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