What is the slope of the normal at the point `("at"^(2), "2 at")` of the parabola `y^(2) = 4ax` ?

Options

A. `(1)/(t)`
B. t
C. `-t`
D. `-(1)/(t)`

Answer

To find the slope of the normal at the point \((a^2, 2a)\) of the parabola given by the equation \(y^2 = 4ax\), we will follow these steps:

### Step 1: Identify the point on the parabola
The point given is \((a^2, 2a)\). Here, \(x_1 = a^2\) and \(y_1 = 2a\).

### Step 2: Find the slope of the tangent
The formula for the slope of the tangent to the parabola \(y^2 = 4ax\) at the point \((x_1, y_1)\) is given by:
\[ \text{slope of tangent} = \frac{y_1}{2a} \]
Substituting \(y_1 = 2a\):
\[ \text{slope of tangent} = \frac{2a}{2a} = 1 \]

### Step 3: Find the slope of the normal
The slope of the normal is the negative reciprocal of the slope of the tangent. Therefore, we can find the slope of the normal using the formula:
\[ \text{slope of normal} = -\frac{1}{\text{slope of tangent}} = -\frac{1}{1} = -1 \]

### Conclusion
Thus, the slope of the normal at the point \((a^2, 2a)\) of the parabola \(y^2 = 4ax\) is \(-1\).

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