`g(x)=asqrt(41-x^(2))`
Function g is defined by the equation above where a is a nonzero real constant. If `g(2i)=sqrt(5)`, where `i=sqrt(-1)`, what is the value of a?

Answer

To solve the problem, we start with the function given by \( g(x) = a \sqrt{41 - x^2} \) and the condition that \( g(2i) = \sqrt{5} \).

### Step 1: Substitute \( x = 2i \) into the function \( g(x) \)

We have:
\[ g(2i) = a \sqrt{41 - (2i)^2} \]

### Step 2: Simplify \( (2i)^2 \)

Calculating \( (2i)^2 \):
\[ (2i)^2 = 4i^2 = 4(-1) = -4 \]

### Step 3: Substitute back into the equation

Now substitute this back into the equation:
\[ g(2i) = a \sqrt{41 - (-4)} = a \sqrt{41 + 4} = a \sqrt{45} \]

### Step 4: Simplify \( \sqrt{45} \)

We can simplify \( \sqrt{45} \):
\[ \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \]

### Step 5: Set up the equation

Now we have:
\[ g(2i) = a \cdot 3\sqrt{5} \]
According to the problem, \( g(2i) = \sqrt{5} \). Therefore, we set up the equation:
\[ a \cdot 3\sqrt{5} = \sqrt{5} \]

### Step 6: Divide both sides by \( \sqrt{5} \)

Assuming \( \sqrt{5} \neq 0 \), we can divide both sides by \( \sqrt{5} \):
\[ a \cdot 3 = 1 \]

### Step 7: Solve for \( a \)

Now, solve for \( a \):
\[ a = \frac{1}{3} \]

Thus, the value of \( a \) is:
\[ \boxed{\frac{1}{3}} \]