A staright line `x/(a)-y/(b)=1` passes through the point (8, 6) and cuts off a triangle of area 12 units from the axes of co-ordinates. Find the equations of the straight line.

Answer

To solve the problem step-by-step, we will follow the instructions given in the video transcript and derive the equations of the straight line.

### Step 1: Substitute the point into the equation of the line
The equation of the line is given as:
\[ \frac{x}{a} - \frac{y}{b} = 1 \]
Since the line passes through the point (8, 6), we substitute \(x = 8\) and \(y = 6\):
\[ \frac{8}{a} - \frac{6}{b} = 1 \]

### Step 2: Rearranging the equation
Rearranging the equation gives:
\[ \frac{8}{a} = 1 + \frac{6}{b} \]
Multiplying through by \(ab\) to eliminate the fractions:
\[ 8b = ab + 6a \]
Rearranging gives:
\[ 8b - ab - 6a = 0 \]
This can be rewritten as:
\[ 8b = ab + 6a \]
From this, we can express \(a\) in terms of \(b\):
\[ a = \frac{8b}{b + 6} \]

### Step 3: Area of the triangle formed by the line
The area \(A\) of the triangle formed by the line and the axes is given as 12 square units. The area of a triangle formed by intercepts on the axes is given by:
\[ A = \frac{1}{2} \times |A| \times |B| \]
where \(A\) and \(B\) are the x-intercept and y-intercept respectively. From the equation of the line, the intercepts are \(A = a\) and \(B = b\). Thus:
\[ \frac{1}{2} \times |a| \times |b| = 12 \]
This simplifies to:
\[ |ab| = 24 \]

### Step 4: Setting up the equations
Now we have two equations:
1. \(ab = 24\)
2. \(a = \frac{8b}{b + 6}\)

### Step 5: Substitute \(a\) in the area equation
Substituting \(a\) from the second equation into the first:
\[ \left(\frac{8b}{b + 6}\right)b = 24 \]
This simplifies to:
\[ 8b^2 = 24(b + 6) \]
Expanding gives:
\[ 8b^2 = 24b + 144 \]
Rearranging gives:
\[ 8b^2 - 24b - 144 = 0 \]
Dividing through by 8:
\[ b^2 - 3b - 18 = 0 \]

### Step 6: Solving the quadratic equation
We can solve this quadratic equation using the factorization method:
\[ (b - 6)(b + 3) = 0 \]
Thus, the solutions for \(b\) are:
\[ b = 6 \quad \text{or} \quad b = -3 \]

### Step 7: Finding corresponding values of \(a\)
1. If \(b = 6\):
\[ a = \frac{8 \cdot 6}{6 + 6} = \frac{48}{12} = 4 \]
2. If \(b = -3\):
\[ a = \frac{8 \cdot (-3)}{-3 + 6} = \frac{-24}{3} = -8 \]

### Step 8: Finding the equations of the lines
Now we can find the equations of the lines for both sets of \(a\) and \(b\):
1. For \(a = 4\) and \(b = 6\):
\[ \frac{x}{4} - \frac{y}{6} = 1 \implies 3x - 2y = 12 \]
2. For \(a = -8\) and \(b = -3\):
\[ \frac{x}{-8} - \frac{y}{-3} = 1 \implies 3x - 8y = -24 \]

### Final Equations of the Straight Line
The equations of the straight line are:
1. \(3x - 2y = 12\)
2. \(3x - 8y = -24\)