Simplify the equation `x^(2)+y^(2)+8x-6y-25=0` to the form `Ax^(2)+By^(2)=K`, by shifting the origin to a suitable point.

Answer

To simplify the equation \( x^2 + y^2 + 8x - 6y - 25 = 0 \) to the form \( Ax^2 + By^2 = K \) by shifting the origin to a suitable point, we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ x^2 + y^2 + 8x - 6y - 25 = 0 \]

### Step 2: Group the terms
Group the \( x \) and \( y \) terms:
\[ (x^2 + 8x) + (y^2 - 6y) - 25 = 0 \]

### Step 3: Complete the square for \( x \)
To complete the square for \( x^2 + 8x \):
1. Take half of the coefficient of \( x \) (which is 8), square it: \( \left(\frac{8}{2}\right)^2 = 16 \).
2. Add and subtract 16 inside the equation:
\[ (x^2 + 8x + 16 - 16) + (y^2 - 6y) - 25 = 0 \]
This simplifies to:
\[ (x + 4)^2 - 16 + (y^2 - 6y) - 25 = 0 \]

### Step 4: Complete the square for \( y \)
To complete the square for \( y^2 - 6y \):
1. Take half of the coefficient of \( y \) (which is -6), square it: \( \left(\frac{-6}{2}\right)^2 = 9 \).
2. Add and subtract 9:
\[ (x + 4)^2 - 16 + (y^2 - 6y + 9 - 9) - 25 = 0 \]
This simplifies to:
\[ (x + 4)^2 - 16 + (y - 3)^2 - 9 - 25 = 0 \]

### Step 5: Combine constant terms
Combine the constant terms:
\[ (x + 4)^2 + (y - 3)^2 - 16 - 9 - 25 = 0 \]
This simplifies to:
\[ (x + 4)^2 + (y - 3)^2 - 50 = 0 \]

### Step 6: Rearrange to the desired form
Rearranging gives:
\[ (x + 4)^2 + (y - 3)^2 = 50 \]

### Step 7: Shift the origin
Now, we can shift the origin to the point \((-4, 3)\) by letting:
\[ X = x + 4 \quad \text{and} \quad Y = y - 3 \]
Thus, the equation becomes:
\[ X^2 + Y^2 = 50 \]

### Final form
This is now in the form \( Ax^2 + By^2 = K \), where \( A = 1 \), \( B = 1 \), and \( K = 50 \).

### Summary
The simplified equation is:
\[ x^2 + y^2 = 50 \]