If the distance of a given point `(alpha,beta)` from each of two straight lines `y = mx` through the origin is d, then `(alpha gamma- beta x)^(2)` is equal to (a) `x^(2) +y^(2)` (b) `d^(2)(x^(2)+y^(2))` (c) `d^(2)` (d) none of these

Options

A. `x^(2) +y^(2)`
B. `d^(2)(x^(2)+y^(2))`
C. `d^(2)`
D. none of these

Explanation

To solve the problem, we need to find the expression for \((\alpha y - \beta x)^2\) given that the distance from the point \((\alpha, \beta)\) to the lines \(y = mx\) is \(d\).

### Step-by-Step Solution:

1. **Understanding the Distance from a Point to a Line**:
The distance \(d\) from a point \((\alpha, \beta)\) to the line \(y = mx\) can be calculated using the formula:
\[ d = \frac{|\beta - m\alpha|}{\sqrt{1 + m^2}} \]

2. **Setting Up the Equation**:
Since the distance \(d\) is given, we can write:
\[ |\beta - m\alpha| = d \sqrt{1 + m^2} \]
Squaring both sides gives:
\[ (\beta - m\alpha)^2 = d^2 (1 + m^2) \]

3. **Substituting \(m\)**:
We know that \(m = \frac{y}{x}\) (the slope of the line). Substituting this into the equation gives:
\[ (\beta - \frac{y}{x}\alpha)^2 = d^2 \left(1 + \left(\frac{y}{x}\right)^2\right) \]

4. **Simplifying the Right Side**:
The right side simplifies to:
\[ d^2 \left(1 + \frac{y^2}{x^2}\right) = d^2 \left(\frac{x^2 + y^2}{x^2}\right) \]
Thus, we have:
\[ (\beta - \frac{y}{x}\alpha)^2 = d^2 \frac{x^2 + y^2}{x^2} \]

5. **Multiplying Both Sides by \(x^2\)**:
To eliminate the fraction, multiply both sides by \(x^2\):
\[ x^2(\beta - \frac{y}{x}\alpha)^2 = d^2 (x^2 + y^2) \]

6. **Rearranging the Left Side**:
The left side can be rewritten as:
\[ (x^2\beta - y\alpha)^2 \]
Therefore, we have:
\[ (x^2\beta - y\alpha)^2 = d^2 (x^2 + y^2) \]

7. **Final Expression**:
We can express this as:
\[ (\alpha y - \beta x)^2 = d^2 (x^2 + y^2) \]

### Conclusion:
Thus, we conclude that:
\[ (\alpha y - \beta x)^2 = d^2 (x^2 + y^2) \]
This corresponds to option (b).