For what values of `a` and `b` the intercepts cut off n the coordinate axes by the line `a x+b y+8=0` are equal in length but opposite in signs to those cut off by the line `2x-3y+6=0` on the axes.

Explanation

To find the values of \( a \) and \( b \) such that the intercepts cut off on the coordinate axes by the line \( ax + by + 8 = 0 \) are equal in length but opposite in sign to those cut off by the line \( 2x - 3y + 6 = 0 \), we will follow these steps:

### Step 1: Determine the intercepts of the line \( 2x - 3y + 6 = 0 \)

To find the x-intercept, set \( y = 0 \):
\[ 2x + 6 = 0 \implies 2x = -6 \implies x = -3 \]
So, the x-intercept is \( -3 \).

To find the y-intercept, set \( x = 0 \):
\[ -3y + 6 = 0 \implies -3y = -6 \implies y = 2 \]
So, the y-intercept is \( 2 \).

### Step 2: Write the intercept form of the line \( ax + by + 8 = 0 \)

Rearranging the equation gives:
\[ ax + by = -8 \]
The intercepts can be found by setting \( y = 0 \) for the x-intercept and \( x = 0 \) for the y-intercept.

For the x-intercept:
\[ ax = -8 \implies x = -\frac{8}{a} \]

For the y-intercept:
\[ by = -8 \implies y = -\frac{8}{b} \]

### Step 3: Set up the equations based on the problem statement

According to the problem, the lengths of the intercepts from the line \( ax + by + 8 = 0 \) must be equal in length but opposite in sign to the intercepts from the line \( 2x - 3y + 6 = 0 \).

Thus, we have:
\[ -\frac{8}{a} = 3 \quad \text{(opposite sign of x-intercept)} \]
\[ -\frac{8}{b} = -2 \quad \text{(opposite sign of y-intercept)} \]

### Step 4: Solve for \( a \) and \( b \)

From the first equation:
\[ -\frac{8}{a} = 3 \implies a = -\frac{8}{3} \]

From the second equation:
\[ -\frac{8}{b} = -2 \implies \frac{8}{b} = 2 \implies b = \frac{8}{2} = 4 \]

### Final Answer
The values of \( a \) and \( b \) are:
\[ a = -\frac{8}{3}, \quad b = 4 \]

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