If area bounded by the curve `y^(2)=4ax` and `y=mx` is `a^(2)//3` , then the value of m, is

Options

A. 1
B. 2
C. 3
D. `sqrt3`

Answer

To solve the problem, we need to find the value of \( m \) such that the area bounded by the curve \( y^2 = 4ax \) and the line \( y = mx \) is equal to \( \frac{a^2}{3} \).

### Step-by-Step Solution:

1. **Identify the curves**:
The first curve is a parabola given by \( y^2 = 4ax \) and the second curve is a straight line given by \( y = mx \).

2. **Find points of intersection**:
To find the points of intersection of the curves, we substitute \( y = mx \) into the equation of the parabola:
\[ (mx)^2 = 4ax \]
This simplifies to:
\[ m^2x^2 - 4ax = 0 \]
Factoring out \( x \):
\[ x(m^2x - 4a) = 0 \]
This gives us \( x = 0 \) (the origin) and \( x = \frac{4a}{m^2} \).

3. **Set up the area integral**:
The area \( A \) bounded by the curves from \( x = 0 \) to \( x = \frac{4a}{m^2} \) can be calculated using the integral:
\[ A = \int_0^{\frac{4a}{m^2}} \left( \sqrt{4ax} - mx \right) \, dx \]

4. **Calculate the integral**:
We first compute the integral:
\[ A = \int_0^{\frac{4a}{m^2}} \sqrt{4ax} \, dx - \int_0^{\frac{4a}{m^2}} mx \, dx \]
The first integral:
\[ \int \sqrt{4ax} \, dx = \frac{2}{3} (4a)^{1/2} x^{3/2} = \frac{2\sqrt{4a}}{3} x^{3/2} \]
Evaluating from \( 0 \) to \( \frac{4a}{m^2} \):
\[ = \frac{2\sqrt{4a}}{3} \left( \frac{4a}{m^2} \right)^{3/2} = \frac{2\sqrt{4a}}{3} \cdot \frac{8a^{3/2}}{m^3} = \frac{16a^2}{3m^3} \]

The second integral:
\[ \int mx \, dx = \frac{m}{2} x^2 \]
Evaluating from \( 0 \) to \( \frac{4a}{m^2} \):
\[ = \frac{m}{2} \left( \frac{4a}{m^2} \right)^2 = \frac{m}{2} \cdot \frac{16a^2}{m^4} = \frac{8a^2}{m^3} \]

5. **Combine the results**:
The total area \( A \) is:
\[ A = \frac{16a^2}{3m^3} - \frac{8a^2}{m^3} = \frac{16a^2 - 24a^2}{3m^3} = \frac{8a^2}{3m^3} \]

6. **Set the area equal to \( \frac{a^2}{3} \)**:
We set the area equal to \( \frac{a^2}{3} \):
\[ \frac{8a^2}{3m^3} = \frac{a^2}{3} \]
Dividing both sides by \( \frac{a^2}{3} \) (assuming \( a \neq 0 \)):
\[ 8 = m^3 \]
Thus, we find:
\[ m = 2 \]

### Final Answer:
The value of \( m \) is \( 2 \).