P(α,β) is a point in first quadrant. If two circles which passes through point P and touches both the coordinate axis, intersect each other orthogonally, then

Options

A. `alpha^(2) + beta^(2) = 4 alpha beta`
B. `(alpha + beta)^(2) = 4 alpha beta`
C. `alpha^(2) +beta^(2) = alpha beta`
D. `alpha^(2) +beta^(2) = 2 alpha beta`

Explanation

To solve the problem, we need to analyze the conditions given for two circles that pass through the point P(α, β) and touch both coordinate axes. Let's go through the solution step-by-step.

### Step 1: Understanding the Circles
The circles touch both the x-axis and y-axis, which means their centers must be at (R, R) where R is the radius of the circles. The general equation of a circle with center (h, k) and radius r is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

For our case, since the circles touch the axes, the centers are at (R, R). Thus, the equation of the circles can be written as:

\[ (x - R)^2 + (y - R)^2 = R^2 \]

### Step 2: Expanding the Circle Equation
Expanding the equation gives:

\[ (x^2 - 2Rx + R^2) + (y^2 - 2Ry + R^2) = R^2 \]

Simplifying this, we have:

\[ x^2 + y^2 - 2Rx - 2Ry + R^2 + R^2 - R^2 = 0 \]

This simplifies to:

\[ x^2 + y^2 - 2R(x + y) + R^2 = 0 \]

### Step 3: Substituting Point P(α, β)
Now, we substitute the coordinates of point P(α, β) into the equation:

\[ \alpha^2 + \beta^2 - 2R(\alpha + \beta) + R^2 = 0 \]

Rearranging gives us:

\[ \alpha^2 + \beta^2 + R^2 = 2R(\alpha + \beta) \]

### Step 4: Finding the Roots
This equation can be treated as a quadratic in R. The general form is:

\[ R^2 - 2R(\alpha + \beta) + (\alpha^2 + \beta^2) = 0 \]

Using the quadratic formula, the roots for R are given by:

\[ R = \frac{2(\alpha + \beta) \pm \sqrt{(2(\alpha + \beta))^2 - 4(\alpha^2 + \beta^2)}}{2} \]

This simplifies to:

\[ R = \alpha + \beta \pm \sqrt{(\alpha + \beta)^2 - (\alpha^2 + \beta^2)} \]

### Step 5: Orthogonality Condition
The circles intersect orthogonally. The condition for two circles to intersect orthogonally is given by:

\[ R_1^2 + R_2^2 = 2R_1R_2 \]

From our earlier steps, we know:

- \(R_1 R_2 = \alpha^2 + \beta^2\)
- \(R_1 + R_2 = 2(\alpha + \beta)\)

Substituting these into the orthogonality condition gives:

\[ 2(\alpha^2 + \beta^2) = 2R_1R_2 \]

### Step 6: Final Equation
From the orthogonality condition, we can derive:

\[ \alpha^2 + \beta^2 = 4\alpha\beta \]

### Conclusion
Thus, the final result is:

\[ \alpha^2 + \beta^2 = 4\alpha\beta \]

This corresponds to **Option 1**.