The area of the region bounded by the curve `(a^4)(y^2)=(2a-x)(x^5)` is to that of the circle whose radius is a, is given by the ratio (a) 4:5 (b) 5 8 (c) 2 3 (d) 3:2.

Options

A. `4:5`
B. `5:8`
C. `2:3`
D. `3:2`

Answer

To find the ratio of the area of the region bounded by the curve \( a^4 y^2 = (2a - x) x^5 \) to the area of a circle with radius \( a \), we can follow these steps:

### Step 1: Analyze the Curve
The equation of the curve is given by:
\[ a^4 y^2 = (2a - x) x^5 \]
To find the points where the curve intersects the x-axis, we set \( y = 0 \):
\[ (2a - x) x^5 = 0 \]
This gives us two points of intersection:
1. \( x = 2a \)
2. \( x = 0 \)

### Step 2: Express \( y \) in Terms of \( x \)
From the equation of the curve, we can express \( y \) as:
\[ y = \sqrt{\frac{(2a - x) x^5}{a^4}} = \frac{\sqrt{(2a - x) x^5}}{a^2} \]

### Step 3: Set Up the Integral for Area
The area \( A_1 \) bounded by the curve can be found using the integral:
\[ A_1 = \int_0^{2a} y \, dx = \int_0^{2a} \frac{\sqrt{(2a - x) x^5}}{a^2} \, dx \]

### Step 4: Change of Variable
To simplify the integral, we can use the substitution \( x = 2a \sin^2 \theta \). Then, we have:
\[ dx = 4a \sin \theta \cos \theta \, d\theta \]
The limits change as follows:
- When \( x = 0 \), \( \theta = 0 \)
- When \( x = 2a \), \( \theta = \frac{\pi}{2} \)

### Step 5: Substitute and Simplify the Integral
Substituting the values into the integral:
\[ A_1 = \int_0^{\frac{\pi}{2}} \frac{\sqrt{(2a - 2a \sin^2 \theta)(2a \sin^2 \theta)^5}}{a^2} \cdot 4a \sin \theta \cos \theta \, d\theta \]
This simplifies to:
\[ A_1 = \int_0^{\frac{\pi}{2}} \frac{\sqrt{2a(1 - \sin^2 \theta)(32a^5 \sin^{10} \theta)}}{a^2} \cdot 4a \sin \theta \cos \theta \, d\theta \]

### Step 6: Evaluate the Integral
After simplification, we can factor out constants and evaluate the integral:
\[ A_1 = \frac{32a^2}{a^2} \int_0^{\frac{\pi}{2}} \sin^6 \theta \cos \theta \, d\theta \]
Using the formula for the integral of powers of sine and cosine, we find:
\[ \int_0^{\frac{\pi}{2}} \sin^m \theta \cos^n \theta \, d\theta = \frac{m! n!}{(m+n+1)!} \]
Applying this formula, we get \( A_1 = \frac{5}{8} \pi a^2 \).

### Step 7: Area of the Circle
The area \( A_2 \) of the circle with radius \( a \) is given by:
\[ A_2 = \pi a^2 \]

### Step 8: Calculate the Ratio
Now, we can find the ratio of the areas:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{\frac{5}{8} \pi a^2}{\pi a^2} = \frac{5}{8} \]

### Final Answer
Thus, the ratio of the area of the region bounded by the curve to the area of the circle is:
\[ \frac{5}{8} \]