The line ax +by+c=0 is an normal to the circle `x^(2)+y^(2)=r^(2)`. The portion of the line ax +by +c=0 intercepted by this circle is of length

Options

A. `sqrtr`
B. r
C. `r^(2)`
D. 2r

Answer

To find the length of the portion of the line \( ax + by + c = 0 \) intercepted by the circle \( x^2 + y^2 = r^2 \), we can follow these steps:

### Step 1: Understand the Circle and the Line
The equation \( x^2 + y^2 = r^2 \) represents a circle centered at the origin (0, 0) with radius \( r \). The line \( ax + by + c = 0 \) is given as a normal to this circle.

**Hint:** A normal line to a circle at a point on the circle passes through the center of the circle.

### Step 2: Determine the Relationship Between the Line and the Circle
Since the line is normal to the circle, it must pass through the center of the circle (which is the origin in this case). Therefore, the distance from the center of the circle to the line must be equal to the radius \( r \).

**Hint:** Use the formula for the distance from a point to a line to find the relationship.

### Step 3: Find the Length of the Intercepted Portion
The line being normal to the circle means it intersects the circle at two points. The length of the portion of the line intercepted by the circle is equal to the length of the diameter of the circle. The diameter \( D \) of a circle is given by the formula:

\[ D = 2r \]

**Hint:** Remember that the diameter is twice the radius.

### Step 4: Conclusion
Thus, the length of the portion of the line \( ax + by + c = 0 \) intercepted by the circle \( x^2 + y^2 = r^2 \) is:

\[ \text{Length} = 2r \]

Therefore, the correct option is \( 2r \).

**Final Answer:** The length of the portion of the line intercepted by the circle is \( 2r \).