When origin is shifted to the point (4, 5) without changing the direction of the coordinate axes, the equation `x^(2)+y^(2)-8x-10y+5=0` is tranformed to the equation `x^(2)+y^(2)=K^(2)`. Value of `|K|` is

Answer

To solve the problem, we need to transform the given equation when the origin is shifted to the point (4, 5). The original equation is:

\[ x^2 + y^2 - 8x - 10y + 5 = 0 \]

### Step 1: Shift the Origin
When the origin is shifted to the point (4, 5), we need to express the new coordinates in terms of the old coordinates. Let:
- \( X = x - 4 \)
- \( Y = y - 5 \)

### Step 2: Substitute the New Coordinates
We will substitute \( x = X + 4 \) and \( y = Y + 5 \) into the original equation:

\[ (X + 4)^2 + (Y + 5)^2 - 8(X + 4) - 10(Y + 5) + 5 = 0 \]

### Step 3: Expand the Equation
Now, we will expand the equation:

1. Expand \( (X + 4)^2 \):
\[ (X + 4)^2 = X^2 + 8X + 16 \]

2. Expand \( (Y + 5)^2 \):
\[ (Y + 5)^2 = Y^2 + 10Y + 25 \]

3. Substitute these expansions into the equation:
\[ X^2 + 8X + 16 + Y^2 + 10Y + 25 - 8X - 32 - 10Y - 50 + 5 = 0 \]

### Step 4: Combine Like Terms
Now, combine all the like terms:

\[ X^2 + Y^2 + (8X - 8X) + (10Y - 10Y) + (16 + 25 - 32 - 50 + 5) = 0 \]

This simplifies to:

\[ X^2 + Y^2 + (16 + 25 - 32 - 50 + 5) = 0 \]

Calculating the constant term:

\[ 16 + 25 = 41 \]
\[ 41 - 32 = 9 \]
\[ 9 - 50 = -41 \]
\[ -41 + 5 = -36 \]

So, we have:

\[ X^2 + Y^2 - 36 = 0 \]

### Step 5: Rearranging the Equation
Rearranging gives us:

\[ X^2 + Y^2 = 36 \]

### Step 6: Relate to the Required Form
The equation \( X^2 + Y^2 = K^2 \) implies:

\[ K^2 = 36 \]

### Step 7: Find |K|
Taking the square root gives:

\[ K = \pm 6 \]

Thus, the absolute value is:

\[ |K| = 6 \]

### Final Answer
The value of \( |K| \) is \( 6 \).

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