Given a right triangle ABC right angled at C and whose legs are given `1+4log_(p^(2))(2p), 1+2^(log_(2)(log_(2)p))` and hypotenuse is given to be `1+log_(2)(4p)`. The are of `DeltaABC` and circle circumscribing it are `Delta_(1) and Delta_(2)` respectively, then
Q. `Delta_(1)+(4Delta_(2))/(pi)` is equal to :

Options

A. 31
B. 28
C. `3+(1)/(sqrt(2))`
D. 199

Answer

To solve the given problem step by step, we will follow the instructions provided in the video transcript and derive the necessary values.

### Step 1: Identify the sides of the triangle
We are given:
- Leg AC = \( 1 + 4 \log_{p^2}(2p) \)
- Leg BC = \( 1 + 2^{\log_2(\log_2 p)} \)
- Hypotenuse AB = \( 1 + \log_2(4p) \)

### Step 2: Simplify Leg AC
Using the property of logarithms:
\[ \log_{p^2}(2p) = \frac{\log_2(2p)}{\log_2(p^2)} = \frac{\log_2(2) + \log_2(p)}{2 \log_2(p)} = \frac{1 + \log_2(p)}{2 \log_2(p)} \]
Thus,
\[ AC = 1 + 4 \cdot \frac{1 + \log_2(p)}{2 \log_2(p)} = 1 + \frac{4 + 4 \log_2(p)}{2 \log_2(p)} = 1 + \frac{2 + 2 \log_2(p)}{\log_2(p)} = \frac{2 + 2 \log_2(p) + \log_2(p)}{\log_2(p)} = \frac{2 + 3 \log_2(p)}{\log_2(p)} \]

### Step 3: Simplify Leg BC
For Leg BC:
\[ BC = 1 + 2^{\log_2(\log_2 p)} = 1 + \log_2 p \]

### Step 4: Simplify Hypotenuse AB
For the hypotenuse:
\[ AB = 1 + \log_2(4p) = 1 + \log_2(4) + \log_2(p) = 1 + 2 + \log_2(p) = 3 + \log_2(p) \]

### Step 5: Calculate the area of triangle ABC (Delta_1)
The area of a right triangle is given by:
\[ \Delta_1 = \frac{1}{2} \times AC \times BC \]
Substituting the values we have:
\[ \Delta_1 = \frac{1}{2} \times \left(1 + 4 \log_{p^2}(2p)\right) \times \left(1 + 2^{\log_2(\log_2 p)}\right) \]
Using the simplified forms:
\[ \Delta_1 = \frac{1}{2} \times \left(\frac{2 + 3 \log_2(p)}{\log_2(p)}\right) \times (1 + \log_2(p)) \]

### Step 6: Calculate the radius of the circumcircle (Delta_2)
The radius \( R \) of the circumcircle of a right triangle is given by:
\[ R = \frac{AB}{2} = \frac{3 + \log_2(p)}{2} \]
Thus, the area of the circumcircle \( \Delta_2 \) is:
\[ \Delta_2 = \pi R^2 = \pi \left(\frac{3 + \log_2(p)}{2}\right)^2 = \frac{\pi (3 + \log_2(p))^2}{4} \]

### Step 7: Calculate \( \Delta_1 + \frac{4\Delta_2}{\pi} \)
Now we need to find:
\[ \Delta_1 + \frac{4\Delta_2}{\pi} \]
Substituting the values we calculated for \( \Delta_1 \) and \( \Delta_2 \):
\[ \Delta_1 + \frac{4}{\pi} \cdot \frac{\pi (3 + \log_2(p))^2}{4} = \Delta_1 + (3 + \log_2(p))^2 \]

### Final Step: Evaluate the expression
After evaluating all the expressions, we can find the final value.