Find the equation of the circle , which passes through the origin and cuts off intercepts `a` and `b` from the axes.

Answer

To find the equation of the circle that passes through the origin and cuts off intercepts \( a \) and \( b \) from the axes, we can follow these steps:

### Step 1: Identify the intercepts
The circle cuts off intercepts \( a \) and \( b \) from the axes. This means that the circle intersects the x-axis at \( (a, 0) \) and the y-axis at \( (0, b) \).

### Step 2: Find the center of the circle
The center of the circle can be found by taking the midpoint of the intercepts. The coordinates of the intercepts are \( (a, 0) \) and \( (0, b) \).

The midpoint \( C \) can be calculated as:
\[ C = \left( \frac{a + 0}{2}, \frac{0 + b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right) \]

### Step 3: Find the radius of the circle
The radius \( R \) of the circle can be calculated using the distance formula between the center \( C \) and one of the intercept points, say \( (a, 0) \):
\[ R = \sqrt{\left( a - \frac{a}{2} \right)^2 + \left( 0 - \frac{b}{2} \right)^2} \]
\[ R = \sqrt{\left( \frac{a}{2} \right)^2 + \left( -\frac{b}{2} \right)^2} = \sqrt{\frac{a^2}{4} + \frac{b^2}{4}} = \frac{1}{2}\sqrt{a^2 + b^2} \]

### Step 4: Write the equation of the circle
The standard equation of a circle with center \( (h, k) \) and radius \( R \) is given by:
\[ (x - h)^2 + (y - k)^2 = R^2 \]
Substituting \( h = \frac{a}{2} \), \( k = \frac{b}{2} \), and \( R = \frac{1}{2}\sqrt{a^2 + b^2} \):
\[ \left( x - \frac{a}{2} \right)^2 + \left( y - \frac{b}{2} \right)^2 = \left( \frac{1}{2}\sqrt{a^2 + b^2} \right)^2 \]
\[ \left( x - \frac{a}{2} \right)^2 + \left( y - \frac{b}{2} \right)^2 = \frac{1}{4}(a^2 + b^2) \]

### Step 5: Expand the equation
Expanding the left-hand side:
\[ \left( x^2 - ax + \frac{a^2}{4} \right) + \left( y^2 - by + \frac{b^2}{4} \right) = \frac{1}{4}(a^2 + b^2) \]
Combining terms:
\[ x^2 + y^2 - ax - by + \frac{a^2}{4} + \frac{b^2}{4} = \frac{1}{4}(a^2 + b^2) \]
Subtracting \( \frac{1}{4}(a^2 + b^2) \) from both sides gives:
\[ x^2 + y^2 - ax - by = 0 \]

### Final Answer
Thus, the required equation of the circle is:
\[ x^2 + y^2 - ax - by = 0 \]