The number of possible straight lines passing through point(2,3) and forming a triangle with coordiante axes whose area is 12 sq. unit is: a. one b. two c. three d. four

Options

A. one
B. two
C. three
D. four

Explanation

To solve the problem of finding the number of possible straight lines passing through the point (2, 3) that form a triangle with the coordinate axes with an area of 12 square units, we can follow these steps:

### Step 1: Understand the area of the triangle
The area \( A \) of a triangle formed by the x-axis, y-axis, and a line can be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base is the x-intercept and the height is the y-intercept of the line.

### Step 2: Set up the area equation
Given that the area is 12 square units, we can set up the equation:
\[ \frac{1}{2} \times x \times y = 12 \]
This simplifies to:
\[ xy = 24 \]

### Step 3: Find the intercepts in terms of the slope
Let the slope of the line be \( m \). The equation of the line passing through the point (2, 3) can be written as:
\[ y - 3 = m(x - 2) \]
Rearranging gives:
\[ y = mx - 2m + 3 \]

To find the x-intercept (where \( y = 0 \)):
\[ 0 = mx - 2m + 3 \implies mx = 2m - 3 \implies x = \frac{2m - 3}{m} \]

To find the y-intercept (where \( x = 0 \)):
\[ y = -2m + 3 \]

### Step 4: Substitute intercepts into the area equation
Now substituting the intercepts into the area equation \( xy = 24 \):
\[ \left(\frac{2m - 3}{m}\right)(-2m + 3) = 24 \]

### Step 5: Simplify the equation
Multiplying out:
\[ (2m - 3)(-2m + 3) = 24m \]
Expanding the left side:
\[ -4m^2 + 6m + 6m - 9 = 24m \]
This simplifies to:
\[ -4m^2 - 12m - 9 = 0 \]
Rearranging gives:
\[ 4m^2 + 12m + 9 = 0 \]

### Step 6: Solve the quadratic equation
We can apply the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = 4, \quad b = 12, \quad c = 9 \]
Calculating the discriminant:
\[ D = b^2 - 4ac = 12^2 - 4 \cdot 4 \cdot 9 = 144 - 144 = 0 \]
Since the discriminant is zero, there is one unique solution for \( m \).

### Step 7: Consider the negative area case
We also need to consider the case where the area could be negative, leading to a second equation:
\[ (2m - 3)(-2m + 3) = -24m \]
Following similar steps as above will yield another quadratic equation.

### Step 8: Solve the second quadratic equation
This will also yield two solutions for \( m \), leading to a total of three distinct lines (one from the first equation and two from the second).

### Conclusion
Thus, the total number of possible straight lines passing through the point (2, 3) that form a triangle with the coordinate axes whose area is 12 square units is:
\[ \text{Three lines} \]

### Final Answer
The correct option is **c. three**.