The circle described on the line joining the points `(0,1), (a, b)` as diameter cuts the x-axis in points whose abscissae are roots of the equation

Options

A. `x^(2)+ax+b=0`
B. `x^(2)-ax+b=0`
C. `x^(2)+ax-b=0`
D. `x^(2)-ax-b=0`

Explanation

To find the equation whose roots are the abscissae of the points where the circle described on the line joining the points (0, 1) and (a, b) as diameter cuts the x-axis, we can follow these steps:

### Step 1: Identify the endpoints of the diameter
The endpoints of the diameter of the circle are given as:
- Point 1: \( (0, 1) \)
- Point 2: \( (a, b) \)

### Step 2: Use the diameter form of the circle's equation
The equation of a circle with endpoints of the diameter at \( (x_1, y_1) \) and \( (x_2, y_2) \) can be expressed as:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]
Substituting the points \( (0, 1) \) and \( (a, b) \):
\[ (x - 0)(x - a) + (y - 1)(y - b) = 0 \]
This simplifies to:
\[ x(x - a) + (y - 1)(y - b) = 0 \]

### Step 3: Substitute \( y = 0 \) to find the points where the circle cuts the x-axis
To find the points where the circle intersects the x-axis, we set \( y = 0 \):
\[ x(x - a) + (0 - 1)(0 - b) = 0 \]
This simplifies to:
\[ x(x - a) + b = 0 \]

### Step 4: Rearrange the equation
Expanding and rearranging the equation gives:
\[ x^2 - ax + b = 0 \]

### Step 5: Identify the quadratic equation
The equation \( x^2 - ax + b = 0 \) is the quadratic equation whose roots are the x-coordinates (abscissae) of the points where the circle intersects the x-axis.

### Conclusion
Thus, the required equation is:
\[ x^2 - ax + b = 0 \]