Find the equation of the parabola whose focus is (5,3) and directrix is the line 3x-4y+1=0.

Answer

To find the equation of the parabola with focus at (5, 3) and directrix given by the line \(3x - 4y + 1 = 0\), we will follow these steps:

### Step 1: Understand the definition of a parabola
A parabola is defined as the set of all points \(P\) such that the distance from \(P\) to the focus \(F\) is equal to the perpendicular distance from \(P\) to the directrix \(D\).

### Step 2: Set up the distance equations
Let \(P(x, y)\) be a point on the parabola. The distance from \(P\) to the focus \(F(5, 3)\) is given by:
\[ PF = \sqrt{(x - 5)^2 + (y - 3)^2} \]

The distance from point \(P\) to the directrix \(3x - 4y + 1 = 0\) can be calculated using the formula for the distance from a point to a line:
\[ PD = \frac{|3x - 4y + 1|}{\sqrt{3^2 + (-4)^2}} = \frac{|3x - 4y + 1|}{5} \]

### Step 3: Set the distances equal
According to the definition of the parabola:
\[ \sqrt{(x - 5)^2 + (y - 3)^2} = \frac{|3x - 4y + 1|}{5} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x - 5)^2 + (y - 3)^2 = \left(\frac{3x - 4y + 1}{5}\right)^2 \]

### Step 5: Simplify both sides
Expanding the left side:
\[ (x - 5)^2 + (y - 3)^2 = (x^2 - 10x + 25) + (y^2 - 6y + 9) = x^2 + y^2 - 10x - 6y + 34 \]

Expanding the right side:
\[ \left(\frac{3x - 4y + 1}{5}\right)^2 = \frac{(3x - 4y + 1)^2}{25} \]
Expanding \((3x - 4y + 1)^2\):
\[ = 9x^2 - 24xy + 16y^2 + 6x - 8y + 1 \]
Thus,
\[ \frac{9x^2 - 24xy + 16y^2 + 6x - 8y + 1}{25} \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 25(x^2 + y^2 - 10x - 6y + 34) = 9x^2 - 24xy + 16y^2 + 6x - 8y + 1 \]

### Step 7: Rearrange the equation
Expanding the left side:
\[ 25x^2 + 25y^2 - 250x - 150y + 850 = 9x^2 - 24xy + 16y^2 + 6x - 8y + 1 \]

Bringing all terms to one side:
\[ 25x^2 - 9x^2 + 25y^2 - 16y^2 + 24xy - 250x - 6x - 150y + 8y + 850 - 1 = 0 \]
This simplifies to:
\[ 16x^2 + 9y^2 + 24xy - 256x - 142y + 849 = 0 \]

### Final Equation
Thus, the equation of the parabola is:
\[ 16x^2 + 9y^2 + 24xy - 256x - 142y + 849 = 0 \]