The line L is given by `2x+by=7`, passes through (8, 3). The line K is parallel to L and has the equation `cx -3y=c`. Find the distance between L and K.

Answer

To solve the problem step by step, we need to find the distance between two parallel lines, L and K, given their equations and a point through which line L passes.

### Step 1: Identify the equations of the lines
The equations of the lines are given as:
- Line L: \( 2x + by = 7 \)
- Line K: \( cx - 3y = c \)

### Step 2: Substitute the point (8, 3) into line L's equation to find b
Since line L passes through the point (8, 3), we can substitute \( x = 8 \) and \( y = 3 \) into the equation of line L:

\[ 2(8) + b(3) = 7 \]

Calculating this gives:

\[ 16 + 3b = 7 \]

### Step 3: Solve for b
Now, we need to isolate \( b \):

\[ 3b = 7 - 16 \]
\[ 3b = -9 \]
\[ b = -3 \]

### Step 4: Write the equation of line L with the found value of b
Now that we have \( b = -3 \), we can write the equation of line L:

\[ 2x - 3y = 7 \]

### Step 5: Determine the value of c for line K
Since line K is parallel to line L, the coefficients of \( x \) and \( y \) must be proportional. The equation of line K is given as \( cx - 3y = c \).

From line L, we have:
- Coefficient of \( x \) is 2
- Coefficient of \( y \) is -3

Thus, for line K to be parallel to line L, we need:

\[ \frac{c}{2} = \frac{-3}{-3} \]

This simplifies to:

\[ \frac{c}{2} = 1 \implies c = 2 \]

### Step 6: Write the equation of line K
Substituting \( c = 2 \) into the equation of line K gives:

\[ 2x - 3y = 2 \]

### Step 7: Find the distance between the two parallel lines
The distance \( d \) between two parallel lines of the form \( Ax + By = C_1 \) and \( Ax + By = C_2 \) is given by the formula:

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

For our lines:
- Line L: \( 2x - 3y = 7 \) (here, \( C_1 = 7 \))
- Line K: \( 2x - 3y = 2 \) (here, \( C_2 = 2 \))

Substituting these values into the distance formula:

\[ d = \frac{|2 - 7|}{\sqrt{2^2 + (-3)^2}} = \frac{|-5|}{\sqrt{4 + 9}} = \frac{5}{\sqrt{13}} \]

### Final Answer
The distance between the two lines L and K is:

\[ \frac{5}{\sqrt{13}} \]

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