Find the equation of parabola whose focus is (4,5) and vertex is (3,6). Also find the length of the latus rectum.

Answer

To find the equation of the parabola with a focus at (4, 5) and a vertex at (3, 6), we can follow these steps:

### Step 1: Determine the orientation of the parabola
The vertex is at (3, 6) and the focus is at (4, 5). Since the focus is to the right of the vertex, the parabola opens to the right.

### Step 2: Find the coordinates of the directrix
The distance between the vertex and the focus is called the focal length, denoted as 'p'. The coordinates of the vertex (h, k) are (3, 6), and the focus (4, 5) gives us the following:
- The focal length \( p = \sqrt{(4 - 3)^2 + (5 - 6)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{2} \).

Since the parabola opens to the right, the directrix will be a vertical line located at \( x = h - p \):
- Directrix: \( x = 3 - \sqrt{2} \).

### Step 3: Write the standard form of the parabola
The standard form of a parabola that opens to the right is given by:
\[ (y - k)^2 = 4p(x - h) \]
Substituting \( h = 3 \), \( k = 6 \), and \( p = \sqrt{2} \):
\[ (y - 6)^2 = 4\sqrt{2}(x - 3) \]

### Step 4: Simplify the equation
Expanding the equation, we have:
\[ (y - 6)^2 = 4\sqrt{2}x - 12\sqrt{2} \]
This is the equation of the parabola.

### Step 5: Find the length of the latus rectum
The length of the latus rectum of a parabola is given by \( 4p \). Since we have \( p = \sqrt{2} \):
\[ \text{Length of latus rectum} = 4\sqrt{2} \]

### Final Answer
The equation of the parabola is:
\[ (y - 6)^2 = 4\sqrt{2}(x - 3) \]
The length of the latus rectum is:
\[ 4\sqrt{2} \]

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