If `sin^(-1)((sqrt(x))/2)+sin^(-1)(sqrt(1-x/4))+tan^(-1)y=(2pi)/3`, then

Options

A. maximum value of `x^(2)+y^(2)` is `(49)/(3)`
B. maximum value of `x^(2)+y^(2)` is 4
C. minimum value of `x^(2)+y^(2)` is `(1)/(3)`
D. minimum value of `x^(2)+y^(2)` is 3

Explanation

To solve the equation \( \sin^{-1}\left(\frac{\sqrt{x}}{2}\right) + \sin^{-1}\left(\sqrt{1 - \frac{x}{4}}\right) + \tan^{-1}(y) = \frac{2\pi}{3} \), we will follow these steps:

### Step 1: Simplify the equation
We can rewrite the equation as:
\[ \sin^{-1}\left(\frac{\sqrt{x}}{2}\right) + \sin^{-1}\left(\sqrt{1 - \frac{x}{4}}\right) + \tan^{-1}(y) = \frac{2\pi}{3} \]

### Step 2: Use the identity for sine and cosine
We know that:
\[ \sin^{-1}(a) + \sin^{-1}(b) + \tan^{-1}(c) = \frac{\pi}{2} \text{ if } a^2 + b^2 + c^2 = 1 \]
Here, we can use the identity:
\[ \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2} \]
Thus, we can express \( \sin^{-1}\left(\frac{\sqrt{x}}{2}\right) + \cos^{-1}\left(\frac{\sqrt{x}}{2}\right) = \frac{\pi}{2} \).

### Step 3: Substitute and simplify
Substituting this into our equation gives:
\[ \frac{\pi}{2} + \tan^{-1}(y) = \frac{2\pi}{3} \]

### Step 4: Isolate \( \tan^{-1}(y) \)
Now, we can isolate \( \tan^{-1}(y) \):
\[ \tan^{-1}(y) = \frac{2\pi}{3} - \frac{\pi}{2} \]
Calculating the right-hand side:
\[ \tan^{-1}(y) = \frac{2\pi}{3} - \frac{3\pi}{6} = \frac{4\pi}{6} - \frac{3\pi}{6} = \frac{\pi}{6} \]

### Step 5: Solve for \( y \)
Taking the tangent of both sides:
\[ y = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \]

### Step 6: Find the values of \( x \)
Now we need to find \( x \) such that:
\[ \sin^{-1}\left(\frac{\sqrt{x}}{2}\right) + \sin^{-1}\left(\sqrt{1 - \frac{x}{4}}\right) = \frac{\pi}{2} \]
This implies:
\[ \frac{\sqrt{x}}{2} = \sqrt{1 - \frac{x}{4}} \]
Squaring both sides:
\[ \frac{x}{4} = 1 - \frac{x}{4} \]
Adding \( \frac{x}{4} \) to both sides:
\[ \frac{x}{4} + \frac{x}{4} = 1 \]
\[ \frac{x}{2} = 1 \implies x = 2 \]

### Step 7: Final values
We have found:
- \( x = 2 \)
- \( y = \frac{1}{\sqrt{3}} \)

### Summary
The values of \( x \) and \( y \) that satisfy the original equation are:
\[ x = 2, \quad y = \frac{1}{\sqrt{3}} \]