If `y(x)` is a solution of differential equation `sqrt(1-x^2) dy/dx + sqrt(1-y^2) = 0` such that `y(1/2) = sqrt3/2`, then

Options

A. (A) `y(1/sqrt2) = -1/sqrt2`
B. (B) `y(1/sqrt2) = sqrt3/2`
C. (C) `y(1/sqrt2) = 1/sqrt2`
D. (D) `y(1/2) = 1/2`

Explanation

To solve the differential equation given by

\[ \sqrt{1-x^2} \frac{dy}{dx} + \sqrt{1-y^2} = 0 \]

with the initial condition \( y\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{2} \), we will follow these steps:

### Step 1: Rearranging the equation
We start by rearranging the differential equation:

\[ \sqrt{1-x^2} \frac{dy}{dx} = -\sqrt{1-y^2} \]

Now, we can separate the variables:

\[ \frac{dy}{\sqrt{1-y^2}} = -\frac{dx}{\sqrt{1-x^2}} \]

### Step 2: Integrating both sides
Next, we integrate both sides:

\[ \int \frac{dy}{\sqrt{1-y^2}} = -\int \frac{dx}{\sqrt{1-x^2}} \]

The integrals can be solved as follows:

\[ \sin^{-1}(y) = -\sin^{-1}(x) + C \]

### Step 3: Solving for the constant \( C \)
To find the constant \( C \), we use the initial condition \( y\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{2} \).

Substituting \( x = \frac{1}{2} \) and \( y = \frac{\sqrt{3}}{2} \):

\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = -\sin^{-1}\left(\frac{1}{2}\right) + C \]

Calculating the values:

\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}, \quad \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

Substituting these values into the equation gives:

\[ \frac{\pi}{3} = -\frac{\pi}{6} + C \]

Adding \(\frac{\pi}{6}\) to both sides:

\[ C = \frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

### Step 4: Final equation
Now we can write the final equation:

\[ \sin^{-1}(y) + \sin^{-1}(x) = \frac{\pi}{2} \]

### Step 5: Finding \( y\left(\frac{1}{\sqrt{2}}\right) \)
We need to find \( y\left(\frac{1}{\sqrt{2}}\right) \). Substitute \( x = \frac{1}{\sqrt{2}} \):

\[ \sin^{-1}(y) + \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{2} \]

Since \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \), we have:

\[ \sin^{-1}(y) + \frac{\pi}{4} = \frac{\pi}{2} \]

Subtracting \(\frac{\pi}{4}\) from both sides:

\[ \sin^{-1}(y) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]

### Step 6: Solving for \( y \)
Taking the sine of both sides:

\[ y = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Conclusion
Thus, the solution to the problem is:

\[ y\left(\frac{1}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} \]