Number of common points to the curves `C_(1){(-1+2cos alpha, 2 sin alpha)}` and `C_(2)(4+3sin theta,3 cos theta)` is/are equal to

Answer

To find the number of common points between the curves \( C_1(-1 + 2\cos \alpha, 2\sin \alpha) \) and \( C_2(4 + 3\sin \theta, 3\cos \theta) \), we will first convert these parametric equations into the standard form of circles.

### Step 1: Identify the parameters for \( C_1 \)

The curve \( C_1 \) is given by the parametric equations:
- \( x = -1 + 2\cos \alpha \)
- \( y = 2\sin \alpha \)

From this, we can identify:
- Center \( (h, k) = (-1, 0) \)
- Radius \( r = 2 \)

### Step 2: Write the equation of circle \( C_1 \)

Using the standard form of a circle's equation \( (x - h)^2 + (y - k)^2 = r^2 \), we substitute the values:
\[ (x + 1)^2 + (y - 0)^2 = 2^2 \]
Thus, the equation of circle \( C_1 \) becomes:
\[ (x + 1)^2 + y^2 = 4 \]

### Step 3: Identify the parameters for \( C_2 \)

The curve \( C_2 \) is given by the parametric equations:
- \( x = 4 + 3\sin \theta \)
- \( y = 3\cos \theta \)

From this, we can identify:
- Center \( (h, k) = (4, 0) \)
- Radius \( r = 3 \)

### Step 4: Write the equation of circle \( C_2 \)

Using the standard form of a circle's equation, we substitute the values:
\[ (x - 4)^2 + (y - 0)^2 = 3^2 \]
Thus, the equation of circle \( C_2 \) becomes:
\[ (x - 4)^2 + y^2 = 9 \]

### Step 5: Find the points of intersection

Now we have two equations:
1. \( (x + 1)^2 + y^2 = 4 \)
2. \( (x - 4)^2 + y^2 = 9 \)

To find the common points, we can eliminate \( y^2 \) by setting the two equations equal to each other.

From the first equation:
\[ y^2 = 4 - (x + 1)^2 \]

Substituting into the second equation:
\[ (x - 4)^2 + 4 - (x + 1)^2 = 9 \]
Expanding and simplifying:
\[ (x - 4)^2 - (x + 1)^2 = 5 \]
\[ (x^2 - 8x + 16) - (x^2 + 2x + 1) = 5 \]
\[ -10x + 15 = 5 \]
\[ -10x = -10 \quad \Rightarrow \quad x = 1 \]

### Step 6: Substitute \( x \) back to find \( y \)

Substituting \( x = 1 \) back into either equation to find \( y \):
Using the first equation:
\[ (1 + 1)^2 + y^2 = 4 \]
\[ 4 + y^2 = 4 \quad \Rightarrow \quad y^2 = 0 \quad \Rightarrow \quad y = 0 \]

### Conclusion

The common point is \( (1, 0) \). Since we found only one point of intersection, the number of common points between the two curves is:

**Answer: 1**