Deriving the Equation of a Parabola with Given Focus and Vertex

The equation of a parabola can be derived using the given focus and vertex. This process involves understanding the geometry of the parabola and the formulas that relate the focus, vertex, and directrix. Let's explore how to derive the equation of a parabola when the focus is at (3, -2) and the vertex is at (3, 4).

Understanding the Geometry of a Parabola

A parabola is a conic section formed by the intersection of a plane and a double-napped cone. The standard form of the equation of a parabola with its vertex at the point (h, k) and focus at the point (h, k p) is given by:

1. Vertex and Focus

Given the vertex (h, k) and the focus (h, k p), the equation of the parabola can be derived as:

[ (x - h)^2 4p(y - k) ]

In this case, ( h 3 ) and ( k 4 ).

2. Distance Between Focus and Vertex

The distance between the focus and the vertex is given by ( p ). Since the focus is at (3, -2) and the vertex is at (3, 4), the distance ( p ) is:

[ p 4 - (-2) 6 ]

3. Equation of the Parabola

Substituting ( h 3 ), ( k 4 ), and ( p 6 ) into the standard equation, we get:

[ (x - 3)^2 4(6)(y - 4) ]

Simplifying the equation, we obtain:

[ (x - 3)^2 24(y - 4) ]

This is the equation of the parabola with the given vertex and focus.

Alternative Method of Deriving the Equation

Another method involves using the distance formula and the definition of a parabola. A parabola is defined as the set of all points equidistant from the focus and the directrix. Here's how we can derive the equation:

1. Axis of Symmetry

Since the x-values of both the focus and vertex are 3, the axis of symmetry is the vertical line ( x 3 ).

2. Directrix

The directrix is a horizontal line above the vertex by the distance ( p ). Since the vertex is 4 and the focus is -2, the distance between them is 6, so the directrix is at ( y 4 6 10 ).

Thus, the equation of the directrix is ( y 10 ).

3. General Point on the Parabola

Let ( (x, y) ) be a general point on the parabola. The distance from this point to the focus (3, -2) must be equal to the distance from this point to the directrix ( y 10 ).

The distance from the point to the focus is:

[ sqrt{(x - 3)^2 (y 2)^2} ]

The distance from the point to the directrix is:

[ |y - 10| ]

4. Equating the Distances

Equating the two distances, we get:

[ sqrt{(x - 3)^2 (y 2)^2} |y - 10| ]

Squaring both sides, we obtain:

[ (x - 3)^2 (y 2)^2 (y - 10)^2 ]

Simplifying, we get:

[ (x - 3)^2 y^2 4y 4 y^2 - 20y 100 ]

Cancelling out ( y^2 ) on both sides, we obtain:

[ (x - 3)^2 4y 4 -20y 100 ]

Rearranging terms, we get:

[ (x - 3)^2 24y - 96 ]

5. Final Equation

Thus, the equation of the parabola is:

[ (x - 3)^2 24(y - 4) ]

Conclusion

The equation of the parabola given the focus at (3, -2) and the vertex at (3, 4) is:

[ (x - 3)^2 24(y - 4) ]

This derivation demonstrates the mathematical steps to find the equation of a parabola using its focus and vertex, highlighting the importance of the geometry and algebraic relationships involved.