If three parabols touch all the lines `x = 0, y = 0` and `x +y =2`, then maximum area of the triangle formed by joining their foci is

Options

A. `sqrt(3)`
B. `sqrt(6)`
C. `(3sqrt(3))/(4)`
D. `(3sqrt(3))/(2)`

Explanation

To solve the problem of finding the maximum area of the triangle formed by joining the foci of three parabolas that touch the lines \(x = 0\), \(y = 0\), and \(x + y = 2\), we can follow these steps:

### Step 1: Identify the Triangle
The lines \(x = 0\), \(y = 0\), and \(x + y = 2\) form a triangle in the first quadrant. The vertices of this triangle can be found by determining the intersection points of these lines.

- The intersection of \(x = 0\) and \(y = 0\) gives the point \(A(0, 0)\).
- The intersection of \(x = 0\) and \(x + y = 2\) gives the point \(B(0, 2)\).
- The intersection of \(y = 0\) and \(x + y = 2\) gives the point \(C(2, 0)\).

Thus, the vertices of the triangle are \(A(0, 0)\), \(B(0, 2)\), and \(C(2, 0)\).

### Step 2: Calculate the Area of Triangle ABC
The area \(A\) of triangle \(ABC\) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take the base as \(BC\) (which is along the line \(y = 0\) from \(B(0, 2)\) to \(C(2, 0)\)), and the height as the distance from point \(A(0, 0)\) to line \(BC\).

The length of \(BC\) can be calculated as:
\[ BC = \sqrt{(2 - 0)^2 + (0 - 2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

The height from point \(A(0, 0)\) to line \(BC\) is simply the y-coordinate of point \(B\) (which is 2).

Thus, the area of triangle \(ABC\) is:
\[ \text{Area} = \frac{1}{2} \times 2\sqrt{2} \times 2 = 2\sqrt{2} \]

### Step 3: Determine the Foci of the Parabolas
Since the parabolas touch the sides of the triangle, their foci will be located at specific points related to the circumcircle of triangle \(ABC\). The circumradius \(R\) of triangle \(ABC\) can be calculated using the formula:
\[ R = \frac{abc}{4A} \]
where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle, and \(A\) is the area of the triangle.

The lengths of the sides can be calculated as follows:
- \(AB = 2\)
- \(AC = 2\)
- \(BC = 2\sqrt{2}\)

Thus,
\[ R = \frac{(2)(2)(2\sqrt{2})}{4 \cdot 2\sqrt{2}} = \frac{8\sqrt{2}}{8\sqrt{2}} = 1 \]

### Step 4: Maximum Area of Triangle Formed by Foci
The maximum area of the triangle formed by the foci of the parabolas inscribed in the circumcircle can be calculated. The maximum area of an equilateral triangle inscribed in a circle is given by:
\[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \]
where \(s\) is the side length of the equilateral triangle. The side length \(s\) can be calculated as \(s = R\sqrt{3}\).

Substituting \(R = 1\):
\[ s = 1\sqrt{3} = \sqrt{3} \]
Thus, the area becomes:
\[ \text{Area} = \frac{\sqrt{3}}{4} (\sqrt{3})^2 = \frac{\sqrt{3}}{4} \cdot 3 = \frac{3\sqrt{3}}{4} \]

### Final Area Calculation
The maximum area of the triangle formed by the foci is:
\[ \text{Area} = \frac{3\sqrt{3}}{2} \]

### Conclusion
The maximum area of the triangle formed by joining the foci of the three parabolas is \(\frac{3\sqrt{3}}{2}\).

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