The length of the common chord of the two circles `x^2+y^2-4y=0` and `x^2+y^2-8x-4y+11=0` is

Options

A. `2sqrt(11)`
B. `3sqrt(2)`
C. `6sqrt(3)`
D. `8sqrt(2)`

Answer

To find the length of the common chord of the two circles given by the equations \(x^2 + y^2 - 4y = 0\) and \(x^2 + y^2 - 8x - 4y + 11 = 0\), we will follow these steps:

### Step 1: Rewrite the equations of the circles in standard form.

1. For the first circle \(C_1: x^2 + y^2 - 4y = 0\):
\[ x^2 + (y^2 - 4y) = 0 \implies x^2 + (y - 2)^2 - 4 = 0 \implies x^2 + (y - 2)^2 = 4 \]
This represents a circle with center at \((0, 2)\) and radius \(r_1 = 2\).

2. For the second circle \(C_2: x^2 + y^2 - 8x - 4y + 11 = 0\):
\[ (x^2 - 8x) + (y^2 - 4y) + 11 = 0 \implies (x - 4)^2 - 16 + (y - 2)^2 - 4 + 11 = 0 \]
\[ \implies (x - 4)^2 + (y - 2)^2 = 9 \]
This represents a circle with center at \((4, 2)\) and radius \(r_2 = 3\).

### Step 2: Find the distance between the centers of the circles.

The centers of the circles are \(C_1(0, 2)\) and \(C_2(4, 2)\). The distance \(d\) between the centers is calculated as follows:
\[ d = \sqrt{(4 - 0)^2 + (2 - 2)^2} = \sqrt{16} = 4 \]

### Step 3: Use the formula for the length of the common chord.

The length \(L\) of the common chord of two intersecting circles can be calculated using the formula:
\[ L = \sqrt{(r_1^2 + r_2^2 - d^2)} \]
where \(r_1\) and \(r_2\) are the radii of the circles and \(d\) is the distance between their centers.

Substituting the values:
- \(r_1 = 2\)
- \(r_2 = 3\)
- \(d = 4\)

We have:
\[ L = \sqrt{(2^2 + 3^2 - 4^2)} = \sqrt{(4 + 9 - 16)} = \sqrt{-3} \]

Since the value under the square root is negative, this indicates that the circles do not intersect, and thus there is no common chord.

### Final Answer:
The length of the common chord is \(0\) since the circles do not intersect.