if the distance between the lines `(x+7y)^2+sqrt2 (x+7y)-42=0` is r, then `(5r^2-10)` equal to

Answer

To solve the problem step-by-step, we will first analyze the given equation and find the distance between the lines represented by it.

### Step 1: Rewrite the Equation
The given equation is:
\[ (x + 7y)^2 + \sqrt{2}(x + 7y) - 42 = 0 \]
Let \( t = x + 7y \). Then, we can rewrite the equation as:
\[ t^2 + \sqrt{2}t - 42 = 0 \]

### Step 2: Solve the Quadratic Equation
We will use the quadratic formula to solve for \( t \):
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = \sqrt{2} \), and \( c = -42 \).

Calculating the discriminant:
\[ b^2 - 4ac = (\sqrt{2})^2 - 4(1)(-42) = 2 + 168 = 170 \]

Now substituting into the quadratic formula:
\[ t = \frac{-\sqrt{2} \pm \sqrt{170}}{2} \]

### Step 3: Find the Two Lines
The two values of \( t \) give us two equations of lines:
1. \( t_1 = \frac{-\sqrt{2} + \sqrt{170}}{2} \)
2. \( t_2 = \frac{-\sqrt{2} - \sqrt{170}}{2} \)

Substituting back for \( t \):
1. \( x + 7y = \frac{-\sqrt{2} + \sqrt{170}}{2} \)
2. \( x + 7y = \frac{-\sqrt{2} - \sqrt{170}}{2} \)

### Step 4: Calculate the Distance Between the Lines
The distance \( r \) between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by:
\[ r = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1 \), \( B = 7 \), \( C_1 = -\frac{-\sqrt{2} + \sqrt{170}}{2} \), and \( C_2 = -\frac{-\sqrt{2} - \sqrt{170}}{2} \).

Calculating \( C_1 - C_2 \):
\[ C_1 - C_2 = \left( \frac{-\sqrt{2} + \sqrt{170}}{2} \right) - \left( \frac{-\sqrt{2} - \sqrt{170}}{2} \right) = \frac{\sqrt{170} + \sqrt{170}}{2} = \frac{2\sqrt{170}}{2} = \sqrt{170} \]

Now calculating \( \sqrt{A^2 + B^2} \):
\[ \sqrt{1^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \]

Thus, the distance \( r \) is:
\[ r = \frac{\sqrt{170}}{5\sqrt{2}} = \frac{\sqrt{170}}{5\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{340}}{10} = \frac{\sqrt{85}}{5} \]

### Step 5: Calculate \( 5r^2 - 10 \)
Now, we need to compute \( 5r^2 - 10 \):
\[ r^2 = \left( \frac{\sqrt{85}}{5} \right)^2 = \frac{85}{25} = \frac{17}{5} \]
Thus,
\[ 5r^2 = 5 \cdot \frac{17}{5} = 17 \]
Finally,
\[ 5r^2 - 10 = 17 - 10 = 7 \]

### Final Answer
The value of \( 5r^2 - 10 \) is:
\[ \boxed{7} \]