if ` sqrt""(1-x^2)+sqrt""(1-y^2)=alpha (x-y) ,`then` (dy)/(dx)=`

Options

A. `sqrt(((1-y^2)/(1-x^2))`
B. `sqrt(((1-x^2)/(1-y^2))`
C. `sqrt((1-x^2)(1-y^2))`
D. None

Answer

To solve the problem, we start with the equation given:

\[ \sqrt{1 - x^2} + \sqrt{1 - y^2} = \alpha (x - y) \]

### Step 1: Substitute Variables
Let's substitute \( x = \sin \theta \) and \( y = \sin \beta \). This gives us:

\[ \sqrt{1 - \sin^2 \theta} + \sqrt{1 - \sin^2 \beta} = \alpha (\sin \theta - \sin \beta) \]

### Step 2: Simplify the Left-Hand Side
Using the identity \( \sqrt{1 - \sin^2 \theta} = \cos \theta \) and \( \sqrt{1 - \sin^2 \beta} = \cos \beta \), we can rewrite the equation as:

\[ \cos \theta + \cos \beta = \alpha (\sin \theta - \sin \beta) \]

### Step 3: Apply the Cosine Addition Formula
Recall the cosine addition formula:

\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]

Applying this to our equation:

\[ 2 \cos\left(\frac{\theta + \beta}{2}\right) \cos\left(\frac{\theta - \beta}{2}\right) = \alpha (\sin \theta - \sin \beta) \]

Using the sine subtraction formula:

\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]

We rewrite the right-hand side:

\[ \alpha \cdot 2 \cos\left(\frac{\theta + \beta}{2}\right) \sin\left(\frac{\theta - \beta}{2}\right) \]

### Step 4: Equate Both Sides
Now we have:

\[ 2 \cos\left(\frac{\theta + \beta}{2}\right) \cos\left(\frac{\theta - \beta}{2}\right) = \alpha \cdot 2 \cos\left(\frac{\theta + \beta}{2}\right) \sin\left(\frac{\theta - \beta}{2}\right) \]

Dividing both sides by \( 2 \cos\left(\frac{\theta + \beta}{2}\right) \) (assuming it is not zero):

\[ \cos\left(\frac{\theta - \beta}{2}\right) = \alpha \sin\left(\frac{\theta - \beta}{2}\right) \]

### Step 5: Solve for \( \theta - \beta \)
Rearranging gives us:

\[ \cot\left(\frac{\theta - \beta}{2}\right) = \alpha \]

Thus, we can express:

\[ \frac{\theta - \beta}{2} = \cot^{-1}(\alpha) \]

### Step 6: Substitute Back
Substituting back for \( \theta \) and \( \beta \):

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

### Step 7: Differentiate Both Sides
Now we differentiate both sides with respect to \( x \):

\[ \frac{1}{2} \left( \frac{1}{\sqrt{1 - x^2}} \cdot \frac{dx}{dx} - \frac{1}{\sqrt{1 - y^2}} \cdot \frac{dy}{dx} \right) = 0 \]

### Step 8: Solve for \( \frac{dy}{dx} \)
Rearranging gives:

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

Thus, we can express \( \frac{dy}{dx} \):

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

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

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