Find the equations of the tangent to the given curve at the indicated point :
`y^(2) = 4ax " at " ((a)/(m^(2)), (2a)/(m))`

Options

A. `y=mx-2am-am^(3)`
B. `m^(3)y=m^(2)x-2am^(2)-a`
C. `m^(3)y=2am^(2)-m^(2)x+a`
D. none of these

Answer

To find the equation of the tangent to the curve \( y^2 = 4ax \) at the point \( \left( \frac{a}{m^2}, \frac{2a}{m} \right) \), we can follow these steps:

### Step 1: Identify the curve and the point
The given curve is \( y^2 = 4ax \) and the point at which we need to find the tangent is \( \left( \frac{a}{m^2}, \frac{2a}{m} \right) \).

### Step 2: Use the formula for the tangent to the parabola
The general equation of the tangent to the parabola \( y^2 = 4ax \) at the point \( (x_1, y_1) \) is given by:
\[ yy_1 = 2a(x + x_1) \]
where \( (x_1, y_1) \) is the point on the parabola.

### Step 3: Substitute the coordinates of the point
In our case, \( x_1 = \frac{a}{m^2} \) and \( y_1 = \frac{2a}{m} \). Substituting these values into the tangent equation:
\[ y \left( \frac{2a}{m} \right) = 2a \left( x + \frac{a}{m^2} \right) \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ \frac{2ay}{m} = 2ax + \frac{2a^2}{m^2} \]

### Step 5: Rearranging the equation
To isolate \( y \), we can multiply through by \( m \):
\[ 2ay = 2amx + \frac{2a^2}{m} \]
Dividing everything by 2:
\[ ay = amx + \frac{a^2}{m} \]

### Step 6: Final equation of the tangent
Rearranging gives us the final equation of the tangent line:
\[ y = mx + \frac{a}{m} \]

### Summary of the solution
The equation of the tangent to the curve \( y^2 = 4ax \) at the point \( \left( \frac{a}{m^2}, \frac{2a}{m} \right) \) is:
\[ y = mx + \frac{a}{m} \]