The line 6x + 8y - 48 intersects the co - ordinate axis at A and B respectively. A line L bisects the area and the perimeter of the traingle OAB, where O is the origin , the number of such lines possible is

Options

A. 1
B. 2
C. 3
D. more than 3

Answer

To solve the problem, we need to find the number of lines that bisect both the area and the perimeter of triangle OAB, where O is the origin, and A and B are the points where the line \(6x + 8y - 48 = 0\) intersects the coordinate axes.

### Step 1: Find the intercepts of the line
The line equation is given as:
\[ 6x + 8y - 48 = 0 \]

To find the x-intercept (point A), set \(y = 0\):
\[ 6x - 48 = 0 \implies x = 8 \]
So, point A is \((8, 0)\).

To find the y-intercept (point B), set \(x = 0\):
\[ 8y - 48 = 0 \implies y = 6 \]
So, point B is \((0, 6)\).

### Step 2: Determine the vertices of triangle OAB
The vertices of triangle OAB are:
- O (the origin): \((0, 0)\)
- A: \((8, 0)\)
- B: \((0, 6)\)

### Step 3: Calculate the area of triangle OAB
The area \(A\) of triangle OAB can be calculated using the formula for the area of a right triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base OA = 8 and the height OB = 6:
\[ \text{Area} = \frac{1}{2} \times 8 \times 6 = 24 \]

### Step 4: Calculate the perimeter of triangle OAB
The lengths of the sides are:
- OA = 8
- OB = 6
- AB (using the distance formula):
\[ AB = \sqrt{(8 - 0)^2 + (0 - 6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

Thus, the perimeter \(P\) is:
\[ P = OA + OB + AB = 8 + 6 + 10 = 24 \]

### Step 5: Find the lines that bisect the area and perimeter
Let the line \(L\) that bisects the area and perimeter of triangle OAB pass through the origin and have the equation \(y = mx\).

To find the point where this line intersects the line \(6x + 8y - 48 = 0\), substitute \(y = mx\) into the line equation:
\[ 6x + 8(mx) - 48 = 0 \implies 6x + 8mx - 48 = 0 \implies (6 + 8m)x = 48 \implies x = \frac{48}{6 + 8m} \]
Then, substituting back to find \(y\):
\[ y = m \left(\frac{48}{6 + 8m}\right) = \frac{48m}{6 + 8m} \]

### Step 6: Calculate the areas of triangles OAD and OBD
The area of triangle OAD is:
\[ \text{Area}_{OAD} = \frac{1}{2} \times OA \times y = \frac{1}{2} \times 8 \times \frac{48m}{6 + 8m} \]
The area of triangle OBD is:
\[ \text{Area}_{OBD} = \frac{1}{2} \times OB \times x = \frac{1}{2} \times 6 \times \frac{48}{6 + 8m} \]

### Step 7: Set the areas equal
To bisect the area:
\[ \text{Area}_{OAD} = \text{Area}_{OBD} \]
This leads to an equation in terms of \(m\). Solving this equation will yield the slope \(m\) of the line \(L\).

### Conclusion
After solving, we find that there is only one line that bisects both the area and the perimeter of triangle OAB.

Thus, the number of such lines possible is:
\[ \boxed{1} \]