The equation of the circle touching the lines y = x at a distance `sqrt(2)` units from the origin is ......

Answer

To find the equation of the circle that touches the lines \( y = x \) at a distance of \( \sqrt{2} \) units from the origin, we can follow these steps:

### Step 1: Understanding the Problem
The circle must touch the lines \( y = x \) and be at a distance of \( \sqrt{2} \) from the origin. The distance from the center of the circle to the line \( y = x \) will be equal to the radius of the circle.

### Step 2: Finding the Center of the Circle
Since the circle touches the line \( y = x \), we can find a point on the line that is \( \sqrt{2} \) units away from the origin. The closest point on the line \( y = x \) to the origin is the point \( (1, 1) \).

### Step 3: Determine the Center
To find the center of the circle, we can take a point that is \( \sqrt{2} \) units away from the origin. The coordinates of the center can be taken as \( (1, 0) \). This point is on the x-axis and is \( 1 \) unit away from the y-axis.

### Step 4: Finding the Radius
The radius of the circle will be the distance from the center \( (1, 0) \) to the line \( y = x \). The distance from a point \( (x_0, y_0) \) to the line \( ax + by + c = 0 \) is given by the formula:

\[ \text{Distance} = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \]

For the line \( y = x \), we can rewrite it as \( x - y = 0 \), where \( a = 1, b = -1, c = 0 \). The distance from the point \( (1, 0) \) to the line is:

\[ \text{Distance} = \frac{|1(1) + (-1)(0) + 0|}{\sqrt{1^2 + (-1)^2}} = \frac{|1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

### Step 5: Setting the Radius
Since the circle must touch the line at a distance of \( \sqrt{2} \) from the origin, we can take the radius as \( 1 \).

### Step 6: Writing 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 \( h = 1, k = 0, \) and \( r = 1 \):

\[ (x - 1)^2 + (y - 0)^2 = 1^2 \]

This simplifies to:

\[ (x - 1)^2 + y^2 = 1 \]

### Final Answer
The equation of the circle is:

\[ (x - 1)^2 + y^2 = 1 \]

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