The equation of the parabloa whose fouce is thepoint `(2,3)` and directirix is the line `x-4y+3=0` is ....... and the lenght of its latus rectum is .....

Answer

To find the equation of the parabola whose focus is at the point \( (2, 3) \) and whose directrix is the line \( x - 4y + 3 = 0 \), we can follow these steps:

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

**Hint:** Remember that the focus is a point, while the directrix is a line.

### Step 2: Find the Equation of the Directrix
We can rearrange the directrix equation to the slope-intercept form to better understand its position:
\[ x - 4y + 3 = 0 \implies 4y = x + 3 \implies y = \frac{1}{4}x + \frac{3}{4} \]
This shows that the directrix has a slope of \( \frac{1}{4} \).

**Hint:** The directrix helps in determining the distance from any point on the parabola.

### Step 3: Use the Definition of a Parabola
The definition of a parabola states that for any point \( P(x, y) \) on the parabola, the distance from \( P \) to the focus \( S \) is equal to the perpendicular distance from \( P \) to the directrix.

### Step 4: Calculate the Distance from \( P \) to the Focus
The distance \( PS \) from point \( P(x, y) \) to the focus \( S(2, 3) \) is given by:
\[ PS = \sqrt{(x - 2)^2 + (y - 3)^2} \]

**Hint:** This is the Euclidean distance formula.

### Step 5: Calculate the Perpendicular Distance from \( P \) to the Directrix
The perpendicular distance \( PM \) from point \( P(x, y) \) to the line \( x - 4y + 3 = 0 \) is given by:
\[ PM = \frac{|x - 4y + 3|}{\sqrt{1^2 + (-4)^2}} = \frac{|x - 4y + 3|}{\sqrt{17}} \]

**Hint:** The formula for the distance from a point to a line can be used here.

### Step 6: Set the Distances Equal
According to the definition of a parabola:
\[ \sqrt{(x - 2)^2 + (y - 3)^2} = \frac{|x - 4y + 3|}{\sqrt{17}} \]

### Step 7: Square Both Sides
Squaring both sides gives:
\[ (x - 2)^2 + (y - 3)^2 = \frac{(x - 4y + 3)^2}{17} \]

### Step 8: Clear the Denominator
Multiply through by 17:
\[ 17((x - 2)^2 + (y - 3)^2) = (x - 4y + 3)^2 \]

### Step 9: Expand Both Sides
Expanding both sides:
\[ 17((x^2 - 4x + 4) + (y^2 - 6y + 9)) = (x^2 - 8xy + 16y^2 + 6x - 24y + 9) \]
This simplifies to:
\[ 17x^2 - 68x + 153 + 17y^2 - 102y = x^2 - 8xy + 16y^2 + 6x - 24y + 9 \]

### Step 10: Rearrange to Form the Parabola Equation
Collect all terms on one side:
\[ 16x^2 + 8xy + y^2 - 74x - 78y + 144 = 0 \]

### Step 11: Find the Length of the Latus Rectum
The length of the latus rectum \( L \) is given by the formula:
\[ L = \frac{2p}{\sqrt{1 + m^2}} \]
where \( p \) is the distance from the focus to the directrix. The distance from the focus \( (2, 3) \) to the line \( x - 4y + 3 = 0 \) can be calculated as:
\[ PM = \frac{|2 - 4(3) + 3|}{\sqrt{17}} = \frac{|2 - 12 + 3|}{\sqrt{17}} = \frac{14}{\sqrt{17}} \]
Thus, the length of the latus rectum is:
\[ L = 2 \times \frac{14}{\sqrt{17}} = \frac{28}{\sqrt{17}} \]

### Final Answers
1. The equation of the parabola is:
\[ 16x^2 + 8xy + y^2 - 74x - 78y + 144 = 0 \]
2. The length of the latus rectum is:
\[ \frac{28}{\sqrt{17}} \]