If the circles described on the line joining the points (0,1) and `(alpha, beta)` as diameter cuts the axis of the points whose abscissae are the roots of the equation `x^(2)-5x+3=0" then "(alpha,beta)=`

Options

A. (5,3)
B. (3,5)
C. (-5,3)
D. (-5,-3)

Answer

To solve the given problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We need to find the coordinates \((\alpha, \beta)\) such that the circle with diameter formed by the points \((0, 1)\) and \((\alpha, \beta)\) intersects the x-axis at the roots of the equation \(x^2 - 5x + 3 = 0\).

### Step 2: Find the Roots of the Quadratic Equation
The roots of the equation \(x^2 - 5x + 3 = 0\) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -5\), and \(c = 3\).

Calculating the discriminant:
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot 3 = 25 - 12 = 13 \]
Now substituting into the quadratic formula:
\[ x = \frac{5 \pm \sqrt{13}}{2} \]
Thus, the roots are:
\[ x_1 = \frac{5 + \sqrt{13}}{2}, \quad x_2 = \frac{5 - \sqrt{13}}{2} \]

### Step 3: Equation of the Circle
The equation of the circle with diameter endpoints \((0, 1)\) and \((\alpha, \beta)\) can be expressed as:
\[ (x - 0)(x - \alpha) + (y - 1)(y - \beta) = 0 \]
This simplifies to:
\[ x(x - \alpha) + (y - 1)(y - \beta) = 0 \]

### Step 4: Substitute \(y = 0\) to Find x-intercepts
To find where the circle intersects the x-axis, set \(y = 0\):
\[ x(x - \alpha) + (0 - 1)(0 - \beta) = 0 \]
This simplifies to:
\[ x^2 - \alpha x + \beta = 0 \]

### Step 5: Relate the Coefficients
From the quadratic equation \(x^2 - \alpha x + \beta = 0\), we can relate the coefficients to the roots we found earlier:
- The sum of the roots \(x_1 + x_2 = \alpha\)
- The product of the roots \(x_1 \cdot x_2 = \beta\)

Using the roots we calculated:
- The sum \(x_1 + x_2 = 5\) (from \(x^2 - 5x + 3 = 0\))
- The product \(x_1 \cdot x_2 = 3\)

### Step 6: Solve for \(\alpha\) and \(\beta\)
From the relationships:
\[ \alpha = 5, \quad \beta = 3 \]

### Final Answer
Thus, the coordinates \((\alpha, \beta)\) are:
\[ (\alpha, \beta) = (5, 3) \]

---