Find the length of the latus rectum of the parabola whose focus is at (2, 3) and directrix is the line `x-4y+3=0` .

Explanation

To find the length of the latus rectum of the parabola with a given focus and directrix, we can follow these steps:

### Step 1: Identify the focus and directrix
The focus of the parabola is given as \( (2, 3) \) and the directrix is given by the equation \( x - 4y + 3 = 0 \).

### Step 2: Write the equation of the directrix in standard form
We can rearrange the directrix equation to the form \( Ax + By + C = 0 \):
\[ x - 4y + 3 = 0 \implies A = 1, B = -4, C = 3 \]

### Step 3: Calculate the perpendicular distance from the focus to the directrix
The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( (x_1, y_1) = (2, 3) \):
\[ d = \frac{|1 \cdot 2 + (-4) \cdot 3 + 3|}{\sqrt{1^2 + (-4)^2}} = \frac{|2 - 12 + 3|}{\sqrt{1 + 16}} = \frac{|2 - 12 + 3|}{\sqrt{17}} = \frac{| -7 |}{\sqrt{17}} = \frac{7}{\sqrt{17}} \]

### Step 4: Relate the distance to the parameter \( a \)
The distance \( d \) from the focus to the directrix is equal to \( 2a \) (where \( a \) is the distance from the vertex to the focus). Thus, we have:
\[ 2a = \frac{7}{\sqrt{17}} \]
To find \( a \), we divide both sides by 2:
\[ a = \frac{7}{2\sqrt{17}} \]

### Step 5: Find the length of the latus rectum
The length of the latus rectum \( L \) of a parabola is given by the formula:
\[ L = 4a \]
Substituting the value of \( a \):
\[ L = 4 \cdot \frac{7}{2\sqrt{17}} = \frac{28}{2\sqrt{17}} = \frac{14}{\sqrt{17}} \]

### Final Answer
The length of the latus rectum of the parabola is \( \frac{14}{\sqrt{17}} \).

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