Find the equation of the parabola with latusrectum joining the points (3,6) and (3,-2).

Answer

To find the equation of the parabola with the latus rectum joining the points (3, 6) and (3, -2), we can follow these steps:

### Step 1: Determine the length of the latus rectum
The points given are (3, 6) and (3, -2). Since the x-coordinates are the same (3), we can find the length of the latus rectum by calculating the distance between the y-coordinates.

\[ \text{Length} = |y_1 - y_2| = |6 - (-2)| = |6 + 2| = 8 \]

### Step 2: Identify the orientation of the parabola
Since the latus rectum is vertical (the x-coordinates are constant), the parabola opens either upwards or downwards. Therefore, the axis of the parabola is parallel to the y-axis.

### Step 3: Find the midpoint of the latus rectum
The midpoint of the latus rectum will serve as the focus of the parabola. We can find the midpoint using the formula:

\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{3 + 3}{2}, \frac{6 + (-2)}{2} \right) = \left( 3, \frac{4}{2} \right) = (3, 2) \]

So, the focus of the parabola is at (3, 2).

### Step 4: Determine the value of \( A \)
The length of the latus rectum is related to \( A \) by the formula \( 4A = \text{Length of latus rectum} \). Thus, we have:

\[ 4A = 8 \implies A = \frac{8}{4} = 2 \]

### Step 5: Write the equation of the parabola
The standard form of the equation of a parabola that opens upwards or downwards is given by:

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

Where \( (h, k) \) is the focus. Here, \( h = 3 \) and \( k = 2 \), and we have \( A = 2 \).

Substituting these values into the equation gives:

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

This simplifies to:

\[ (y - 2)^2 = 8(x - 3) \]

### Final Answer
The equation of the parabola is:

\[ (y - 2)^2 = 8(x - 3) \]

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