The points representing ` 3 sqrt(sqrt""5 + isqrt""3)` lie

Options

A. on a circle centre (0,0) and radius `2 sqrt""2`
B. on a straight line
C. on a circle centre (0,0) and radius `sqrt""2 `
D. none of these

Answer

To solve the problem of finding the points representing \( 3 \sqrt{\sqrt{5} + i\sqrt{3}} \), we will follow these steps:

### Step 1: Simplify the expression inside the square root
We start with the expression \( \sqrt{5} + i\sqrt{3} \).

### Step 2: Find the magnitude of the complex number
The magnitude \( |z| \) of a complex number \( z = a + bi \) is given by:
\[ |z| = \sqrt{a^2 + b^2} \]
For our complex number \( \sqrt{5} + i\sqrt{3} \):
- \( a = \sqrt{5} \)
- \( b = \sqrt{3} \)

Calculating the magnitude:
\[ |z| = \sqrt{(\sqrt{5})^2 + (\sqrt{3})^2} = \sqrt{5 + 3} = \sqrt{8} = 2\sqrt{2} \]

### Step 3: Find the argument of the complex number
The argument \( \theta \) of a complex number \( z = a + bi \) is given by:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
For our complex number:
\[ \theta = \tan^{-1}\left(\frac{\sqrt{3}}{\sqrt{5}}\right) \]

### Step 4: Write the complex number in polar form
The polar form of a complex number is given by:
\[ z = r(\cos \theta + i \sin \theta) \]
Where \( r \) is the magnitude and \( \theta \) is the argument. Thus, we can express \( \sqrt{5} + i\sqrt{3} \) as:
\[ \sqrt{5} + i\sqrt{3} = 2\sqrt{2} \left( \cos\left(\tan^{-1}\left(\frac{\sqrt{3}}{\sqrt{5}}\right)\right) + i \sin\left(\tan^{-1}\left(\frac{\sqrt{3}}{\sqrt{5}}\right)\right) \right) \]

### Step 5: Take the square root of the complex number
To find \( \sqrt{\sqrt{5} + i\sqrt{3}} \), we take the square root in polar form:
\[ \sqrt{z} = \sqrt{r} \left( \cos\left(\frac{\theta}{2}\right) + i \sin\left(\frac{\theta}{2}\right) \right) \]
Calculating \( \sqrt{r} \):
\[ \sqrt{r} = \sqrt{2\sqrt{2}} = 2^{3/4} = 2^{0.75} \]
Calculating \( \frac{\theta}{2} \):
\[ \frac{\theta}{2} = \frac{1}{2} \tan^{-1}\left(\frac{\sqrt{3}}{\sqrt{5}}\right) \]

### Step 6: Multiply by 3
Now we multiply the result by 3:
\[ 3 \sqrt{\sqrt{5} + i\sqrt{3}} = 3 \cdot 2^{0.75} \left( \cos\left(\frac{1}{2} \tan^{-1}\left(\frac{\sqrt{3}}{\sqrt{5}}\right)\right) + i \sin\left(\frac{1}{2} \tan^{-1}\left(\frac{\sqrt{3}}{\sqrt{5}}\right)\right) \right) \]

### Final Result
The points representing \( 3 \sqrt{\sqrt{5} + i\sqrt{3}} \) can be expressed in polar form as above.