If ` A(1,p^(2)), B(0,1) and C(p,0) ` are the coordinates of three points then the value of p for which the area of the triangle ABC is minimum is

Options

A. ` (1)/(sqrt3)`
B. ` -(1)/(sqrt3)`
C. ` (1)/(sqrt3) or -(1)/(sqrt3) `
D. none of these

Answer

To find the value of \( p \) for which the area of triangle \( ABC \) is minimum, we can follow these steps:

### Step 1: Identify the coordinates of the points
The coordinates of the points are:
- \( A(1, p^2) \)
- \( B(0, 1) \)
- \( C(p, 0) \)

### Step 2: Use the formula for the area of a triangle
The area \( A \) of triangle \( ABC \) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates into the formula:
\[ \text{Area} = \frac{1}{2} \left| 1(1 - 0) + 0(0 - p^2) + p(p^2 - 1) \right| \]

### Step 3: Simplify the expression
Calculating the expression:
\[ = \frac{1}{2} \left| 1 + p(p^2 - 1) \right| \]
\[ = \frac{1}{2} \left| 1 + p^3 - p \right| \]
\[ = \frac{1}{2} \left| p^3 - p + 1 \right| \]

### Step 4: Set the area expression to zero for minimum area
For the area to be minimum, we set the expression inside the absolute value to zero:
\[ p^3 - p + 1 = 0 \]

### Step 5: Solve the cubic equation
To find the roots of the cubic equation \( p^3 - p + 1 = 0 \), we can use numerical methods or graphing techniques, as it may not have simple rational roots.

### Step 6: Analyze the function
We can analyze the function \( f(p) = p^3 - p + 1 \) to find the critical points. Taking the derivative:
\[ f'(p) = 3p^2 - 1 \]
Setting the derivative to zero:
\[ 3p^2 - 1 = 0 \implies p^2 = \frac{1}{3} \implies p = \pm \frac{1}{\sqrt{3}} \]

### Step 7: Determine the minimum area
We can evaluate the function \( f(p) \) at \( p = \frac{1}{\sqrt{3}} \) and \( p = -\frac{1}{\sqrt{3}} \) to find the minimum area.

### Step 8: Conclusion
The value of \( p \) for which the area of triangle \( ABC \) is minimum is:
\[ p = -\frac{1}{\sqrt{3}} \text{ or } p = \frac{1}{\sqrt{3}} \]