What conic does the equation
`25(x^2+y^2-2x+1)=(4x-3y+1)^2` represent?

Answer

To determine what conic section the given equation represents, we will start by simplifying the equation step by step.

### Step 1: Rewrite the given equation
The given equation is:
\[ 25(x^2 + y^2 - 2x + 1) = (4x - 3y + 1)^2 \]

### Step 2: Move all terms to one side
We can rewrite the equation as:
\[ 25(x^2 + y^2 - 2x + 1) - (4x - 3y + 1)^2 = 0 \]

### Step 3: Expand both sides
Now, let's expand both sides of the equation. Start with the left side:
- Expand \( 25(x^2 + y^2 - 2x + 1) \):
\[ 25x^2 + 25y^2 - 50x + 25 \]

- Expand the right side \( (4x - 3y + 1)^2 \):
\[ (4x - 3y + 1)(4x - 3y + 1) = 16x^2 - 24xy + 8x + 9y^2 - 6y + 1 \]

### Step 4: Combine the expanded forms
Now, we can combine both expansions:
\[ 25x^2 + 25y^2 - 50x + 25 - (16x^2 - 24xy + 8x + 9y^2 - 6y + 1) = 0 \]

### Step 5: Simplify the equation
Combine like terms:
- For \( x^2 \): \( 25x^2 - 16x^2 = 9x^2 \)
- For \( y^2 \): \( 25y^2 - 9y^2 = 16y^2 \)
- For \( x \): \( -50x - 8x = -58x \)
- For \( y \): \( 6y \)
- Constant terms: \( 25 - 1 = 24 \)

So, we have:
\[ 9x^2 + 16y^2 - 58x + 6y + 24 = 0 \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ 9x^2 + 16y^2 - 58x + 6y = -24 \]

### Step 7: Completing the square
Now, we will complete the square for the \( x \) and \( y \) terms.

For \( x \):
\[ 9(x^2 - \frac{58}{9}x) \]
To complete the square, take half of \(-\frac{58}{9}\), square it, and add/subtract it inside the bracket.

Calculating:
\[ \left(\frac{29}{9}\right)^2 = \frac{841}{81} \]
Thus,
\[ 9\left(x^2 - \frac{58}{9}x + \frac{841}{81} - \frac{841}{81}\right) = 9\left((x - \frac{29}{9})^2 - \frac{841}{81}\right) \]

For \( y \):
\[ 16(y^2 + \frac{3}{8}y) \]
Completing the square:
\[ \left(\frac{3}{16}\right)^2 = \frac{9}{256} \]
Thus,
\[ 16\left(y^2 + \frac{3}{8}y + \frac{9}{256} - \frac{9}{256}\right) = 16\left((y + \frac{3}{16})^2 - \frac{9}{256}\right) \]

### Step 8: Substitute back into the equation
Substituting back into the equation gives:
\[ 9\left(x - \frac{29}{9}\right)^2 + 16\left(y + \frac{3}{16}\right)^2 = C \]
Where \( C \) is a constant derived from the completed squares.

### Step 9: Identify the conic section
The resulting equation is in the form of a conic section. Since we have both \( x^2 \) and \( y^2 \) terms with different coefficients, this indicates that the conic is an **ellipse**.

### Conclusion
The equation represents an **ellipse**.

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