The directrix of a conic section is the straight line `3x-4y+5-0` and the focus is `(2,3)`. If the eccontricity e is 1, find the equation to the coin section. Is the coin sction a parabola?

Answer

To find the equation of the conic section given the directrix and focus, we will follow these steps:

### Step 1: Identify the given information
- Directrix: \(3x - 4y + 5 = 0\)
- Focus: \(F(2, 3)\)
- Eccentricity \(e = 1\)

### Step 2: Understand the definition of a parabola
A parabola is defined as the locus of points where the distance from a point (focus) is equal to the distance from a line (directrix).

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

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

### Step 4: Set the distances equal
Since \(e = 1\), we have:
\[ PF = PM \]
Thus,
\[ \sqrt{(x - 2)^2 + (y - 3)^2} = \frac{|3x - 4y + 5|}{5} \]

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

### Step 6: Expand both sides
Expanding the left side:
\[ (x - 2)^2 + (y - 3)^2 = (x^2 - 4x + 4) + (y^2 - 6y + 9) = x^2 + y^2 - 4x - 6y + 13 \]

Expanding the right side:
\[ \left(\frac{3x - 4y + 5}{5}\right)^2 = \frac{(3x - 4y + 5)^2}{25} \]
\[ = \frac{(9x^2 - 24xy + 16y^2 + 30x - 40y + 25)}{25} \]

### Step 7: Clear the fraction by multiplying through by 25
Multiplying both sides by 25:
\[ 25(x^2 + y^2 - 4x - 6y + 13) = 9x^2 - 24xy + 16y^2 + 30x - 40y + 25 \]

### Step 8: Rearrange the equation
Expanding the left side:
\[ 25x^2 + 25y^2 - 100x - 150y + 325 = 9x^2 - 24xy + 16y^2 + 30x - 40y + 25 \]

Rearranging gives:
\[ 25x^2 - 9x^2 + 25y^2 - 16y^2 + 24xy - 100x - 30x - 150y + 40y + 325 - 25 = 0 \]
\[ 16x^2 + 9y^2 + 24xy - 130x - 110y + 300 = 0 \]

### Step 9: Final equation of the parabola
The equation of the parabola is:
\[ 16x^2 + 9y^2 + 24xy - 130x - 110y + 300 = 0 \]

### Step 10: Conclusion
Since the eccentricity \(e = 1\), we confirm that this conic section is indeed a parabola.

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