In the right triangle `BAC, angle A=pi/2 and a+b=8`. The area of the triangle is maximum when `angle C` , is

Options

A. `pi//3`
B. `pi//4`
C. `pi//6`
D. `pi//2`

Answer

To solve the problem, we need to maximize the area of the right triangle \( BAC \) with the given conditions. Let's go through the steps systematically.

### Step 1: Understanding the triangle and the given conditions
We have a right triangle \( BAC \) where \( \angle A = \frac{\pi}{2} \) (90 degrees). We are given that \( a + b = 8 \), where \( a \) and \( b \) are the lengths of the sides opposite angles \( A \) and \( B \) respectively.

### Step 2: Express the area of the triangle
The area \( A \) of triangle \( BAC \) can be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times b \times c \]
However, we need to express the area in terms of a single variable.

### Step 3: Relate \( b \) and \( c \) using the given condition
From the condition \( a + b = 8 \), we can express \( b \) in terms of \( a \):
\[ b = 8 - a \]

### Step 4: Use the Pythagorean theorem
Using the Pythagorean theorem, we have:
\[ c^2 = a^2 + b^2 \]
Substituting for \( b \):
\[ c^2 = a^2 + (8 - a)^2 \]
Expanding this:
\[ c^2 = a^2 + (64 - 16a + a^2) = 2a^2 - 16a + 64 \]

### Step 5: Express \( c \) in terms of \( a \)
Taking the square root:
\[ c = \sqrt{2a^2 - 16a + 64} \]

### Step 6: Substitute \( b \) and \( c \) into the area formula
Now substituting \( b \) and \( c \) into the area formula:
\[ A = \frac{1}{2} \times (8 - a) \times \sqrt{2a^2 - 16a + 64} \]

### Step 7: Differentiate the area with respect to \( a \)
To find the maximum area, we differentiate \( A \) with respect to \( a \) and set the derivative to zero:
\[ \frac{dA}{da} = 0 \]
This requires using the product rule and chain rule. After differentiating and simplifying, we will find the critical points.

### Step 8: Solve for \( a \)
Setting the derivative equal to zero will yield a value for \( a \). Solving this will give us the value of \( a \) at which the area is maximized.

### Step 9: Find \( b \) and \( c \)
Once we have \( a \), we can find \( b \) using \( b = 8 - a \) and then substitute \( a \) back into the equation for \( c \).

### Step 10: Determine \( \angle C \)
Finally, we can find \( \sin C \) using the relation:
\[ \sin C = \frac{c}{a} \]
From this, we can find \( \angle C \) in radians.

### Conclusion
After performing all calculations, we will find that the angle \( C \) for which the area of the triangle is maximized is:
\[ C = \frac{\pi}{3} \text{ (or 60 degrees)} \]