Question

Write the equation of the parabola with vertex (3,2)
and focus (7,2).

Answer 1

To write the equation of a parabola, we need to know its vertex and focus. Given the vertex at \((3, 2)\) and the focus at \((7, 2)\), we can determine that the parabola opens horizontally to the right because the focus is to the right of the vertex.

The general form of a horizontally opening parabola is given by the equation:

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

where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus.

1. The vertex \((h, k) = (3, 2)\).
2. The distance \(p\) is the horizontal distance from the vertex to the focus. The focus at \((7, 2)\) means:

\[
p = 7 - 3 = 4
\]

Now we can substitute \(h\), \(k\), and \(p\) into the equation:

\[
(y - 2)^2 = 4 \cdot 4 (x - 3)
\]

This simplifies to:

\[
(y - 2)^2 = 16(x - 3)
\]

So, the equation of the parabola is:

\[
(y - 2)^2 = 16(x - 3)
\]