The length of the common chord of the ellipse `((x-1)^2)/9+((y-2)^2)/4=1` and the circle `(x-1)^2+(y-2)^2=1` is (A) 2 (B) `sqrt3` (C) 4 (D) none of these

Options

A. zero
B. one
C. three
D. eight

Answer

To find the length of the common chord of the given ellipse and circle, we will follow these steps:

### Step 1: Write down the equations of the ellipse and the circle.
The equations given are:
1. Ellipse: \(\frac{(x-1)^2}{9} + \frac{(y-2)^2}{4} = 1\)
2. Circle: \((x-1)^2 + (y-2)^2 = 1\)

### Step 2: Substitute variables for simplification.
Let:
- \(a = x - 1\)
- \(b = y - 2\)

Then, we can rewrite the equations as:
1. Ellipse: \(\frac{a^2}{9} + \frac{b^2}{4} = 1\)
2. Circle: \(a^2 + b^2 = 1\)

### Step 3: Express \(a^2\) in terms of \(b^2\) from the circle's equation.
From the circle's equation:
\[ a^2 = 1 - b^2 \]

### Step 4: Substitute \(a^2\) into the ellipse's equation.
Substituting \(a^2\) into the ellipse's equation gives:
\[ \frac{1 - b^2}{9} + \frac{b^2}{4} = 1 \]

### Step 5: Clear the fractions by finding a common denominator.
The common denominator for 9 and 4 is 36. Multiply the entire equation by 36:
\[ 36 \left(\frac{1 - b^2}{9}\right) + 36 \left(\frac{b^2}{4}\right) = 36 \]
This simplifies to:
\[ 4(1 - b^2) + 9b^2 = 36 \]

### Step 6: Expand and simplify the equation.
Expanding gives:
\[ 4 - 4b^2 + 9b^2 = 36 \]
Combine like terms:
\[ 4 + 5b^2 = 36 \]

### Step 7: Solve for \(b^2\).
Rearranging gives:
\[ 5b^2 = 36 - 4 \]
\[ 5b^2 = 32 \]
\[ b^2 = \frac{32}{5} \]

### Step 8: Substitute \(b^2\) back to find \(a^2\).
Now substitute \(b^2\) back into the equation for \(a^2\):
\[ a^2 = 1 - b^2 = 1 - \frac{32}{5} = \frac{5 - 32}{5} = \frac{-27}{5} \]

### Step 9: Analyze the results.
Since \(a^2\) is negative (\(-\frac{27}{5}\)), it indicates that there are no real solutions for \(a\) and \(b\). This means that the ellipse and the circle do not intersect.

### Conclusion:
Since the curves do not intersect, the length of the common chord is \(0\).

### Final Answer:
The length of the common chord is \(0\), which corresponds to option (D) none of these.

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