Normal to the circle `x^(2)+y^(2)=4` divides the circle having centre at (2,4) and radius 2 in the ares of ratio `(pi -2) : (3pi +2)`. Then the normal can be

Explanation

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Understand the Problem
We have a circle given by the equation \( x^2 + y^2 = 4 \), which has a center at the origin (0, 0) and a radius of 2. We also have another circle with center (2, 4) and radius 2. A normal to the first circle divides the area of the second circle in the ratio \( \frac{\pi - 2}{3\pi + 2} \).

### Step 2: Write the Equation of the Normal
The normal to the circle at any point can be expressed in the form \( y = mx \) where \( m \) is the slope. Since the normal passes through the origin, we can represent it as such.

### Step 3: Find the Area of the Segments
The area of the segment cut off by the normal in the second circle can be calculated using the formula for the area of a circular segment. The area \( A \) of a segment is given by:
\[ A = \frac{r^2}{2}(\theta - \sin \theta) \]
where \( r \) is the radius and \( \theta \) is the angle in radians.

### Step 4: Set Up the Areas
Let \( A_1 \) be the area of the segment on one side of the normal and \( A_2 \) be the area on the other side. The given ratio is:
\[ \frac{A_1}{A_2} = \frac{\pi - 2}{3\pi + 2} \]
Using the area formula, we can express \( A_1 \) and \( A_2 \) in terms of \( \theta \) (the angle subtended by the segment at the center of the second circle).

### Step 5: Calculate Areas
The total area of the second circle is:
\[ \text{Total Area} = \pi r^2 = \pi \cdot 2^2 = 4\pi \]
Thus, we can express \( A_2 \) as:
\[ A_2 = 4\pi - A_1 \]

### Step 6: Substitute in the Ratio
Substituting \( A_2 \) into the ratio gives:
\[ \frac{A_1}{4\pi - A_1} = \frac{\pi - 2}{3\pi + 2} \]
Cross-multiplying leads to:
\[ A_1(3\pi + 2) = (4\pi - A_1)(\pi - 2) \]

### Step 7: Solve for \( A_1 \)
Expanding and simplifying the equation will allow us to find \( A_1 \) in terms of \( \theta \).

### Step 8: Find \( \theta \)
After simplifying, we will find that \( \theta \) satisfies the equation:
\[ \theta - \sin \theta = \frac{\pi}{2} - 1 \]
By inspection or numerical methods, we find that \( \theta = \frac{\pi}{2} \) satisfies this equation.

### Step 9: Calculate the Lengths
Using \( \theta = \frac{\pi}{2} \), we can find the lengths involved using the Pythagorean theorem.

### Step 10: Find the Slope \( m \)
Using the distance from the center of the second circle (2, 4) to the line \( y = mx \), we set up the equation:
\[ \frac{|2m - 4|}{\sqrt{m^2 + 1}} = \sqrt{2} \]
Squaring both sides and simplifying will yield a quadratic in \( m \).

### Step 11: Solve the Quadratic
The quadratic will yield two possible values for \( m \), which correspond to the slopes of the normals.

### Final Answer
The equations of the normals are:
\[ y = x \quad \text{and} \quad y = 7x \]