Number of common chords of a parabola & a circle can be

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the problem of finding the number of common chords between a parabola and a circle, we will follow these steps:

### Step 1: Understand the Intersection Points
A parabola can intersect a circle at a maximum of 4 points. This is because the degree of the polynomial formed when we set the equations equal to each other can be at most 4.

**Hint:** Visualize the parabola and circle to understand how they can intersect at multiple points.

### Step 2: Set Up the Equations
Let's consider the parabola given by the equation \( y = \frac{x^2}{4a} \) and a circle represented by the general equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \).

**Hint:** Write down the equations clearly to avoid confusion during substitution.

### Step 3: Substitute to Find Intersection Points
Substituting \( y = \frac{x^2}{4a} \) into the circle's equation allows us to find the intersection points. This leads to a polynomial equation in \( x \).

**Hint:** Ensure you substitute correctly and simplify the equation step by step.

### Step 4: Determine the Degree of the Resulting Equation
After substitution, the resulting equation will be of degree 4 (the highest power of \( x \)). This indicates that there can be up to 4 intersection points.

**Hint:** Remember that the degree of the polynomial indicates the maximum number of solutions (intersection points).

### Step 5: Calculate the Number of Common Chords
To find the number of common chords, we need to select pairs of intersection points. The number of ways to choose 2 points from 4 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points (4) and \( r \) is the number of points to choose (2).

**Hint:** Use the combination formula \( \binom{4}{2} = \frac{4!}{2!(4-2)!} \).

### Step 6: Perform the Calculation
Calculating \( \binom{4}{2} \):
\[ \binom{4}{2} = \frac{4!}{2! \cdot 2!} = \frac{4 \times 3}{2 \times 1} = 6 \]

**Hint:** Break down the factorials to simplify your calculations.

### Conclusion
The maximum number of common chords between a parabola and a circle is **6**.

**Final Answer:** 6