If points `(a^2, 0),(0, b^2) and (1, 1)` are collinear, then

Options

A. `(1)/(a^(2))+(1)/(b^(2))=1`
B. `(1)/(a)+(1)/(b)=1`
C. `a^(2)+b^(2)=1`
D. none of these

Explanation

To determine the relationship between the points \((a^2, 0)\), \((0, b^2)\), and \((1, 1)\) being collinear, we can use the formula for the area of a triangle formed by three points in the Cartesian coordinate system. The area of the triangle formed by the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Since the points are collinear, the area must be equal to 0.

### Step 1: Set up the points
Let:
- \( (x_1, y_1) = (a^2, 0) \)
- \( (x_2, y_2) = (0, b^2) \)
- \( (x_3, y_3) = (1, 1) \)

### Step 2: Substitute the points into the area formula
Substituting the coordinates into the area formula gives us:

\[ \text{Area} = \frac{1}{2} \left| a^2(b^2 - 1) + 0(1 - 0) + 1(0 - b^2) \right| \]

This simplifies to:

\[ \text{Area} = \frac{1}{2} \left| a^2(b^2 - 1) - b^2 \right| \]

### Step 3: Set the area to zero
Since the points are collinear, we set the area equal to 0:

\[ \frac{1}{2} \left| a^2(b^2 - 1) - b^2 \right| = 0 \]

This implies:

\[ \left| a^2(b^2 - 1) - b^2 \right| = 0 \]

Thus, we have:

\[ a^2(b^2 - 1) - b^2 = 0 \]

### Step 4: Rearranging the equation
Rearranging gives us:

\[ a^2(b^2 - 1) = b^2 \]

### Step 5: Divide both sides by \(b^2\) (assuming \(b \neq 0\))
Dividing both sides by \(b^2\) gives:

\[ a^2 \left(1 - \frac{1}{b^2}\right) = 1 \]

### Step 6: Rearranging again
Rearranging this gives:

\[ a^2 = \frac{b^2}{b^2 - 1} \]

### Step 7: Final expression
This can also be expressed as:

\[ \frac{1}{a^2} + \frac{1}{b^2} = 1 \]

This is the relationship we were looking for.