Find the equation of normal to the curve `x = at^(2), y=2at` at point 't'.

Answer

To find the equation of the normal to the curve defined by the parametric equations \( x = at^2 \) and \( y = 2at \) at a point corresponding to the parameter \( t \), we can follow these steps:

### Step 1: Identify the point on the curve
At a given parameter \( t \), the coordinates of the point on the curve can be found as follows:
- \( x_1 = at^2 \)
- \( y_1 = 2at \)

### Step 2: Find the derivatives
To find the slope of the tangent line at the point, we need to compute \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \):
- \( \frac{dy}{dt} = 2a \)
- \( \frac{dx}{dt} = 2at \)

### Step 3: Calculate the slope of the tangent
The slope of the tangent line \( m_t \) can be calculated using the formula:
\[ m_t = \frac{dy/dt}{dx/dt} = \frac{2a}{2at} = \frac{1}{t} \]

### Step 4: Determine the slope of the normal
The slope of the normal line \( m_n \) is the negative reciprocal of the slope of the tangent:
\[ m_n = -\frac{1}{m_t} = -t \]

### Step 5: Write the equation of the normal
The equation of the normal line can be expressed in point-slope form:
\[ y - y_1 = m_n (x - x_1) \]
Substituting \( x_1 = at^2 \), \( y_1 = 2at \), and \( m_n = -t \):
\[ y - 2at = -t(x - at^2) \]

### Step 6: Rearranging the equation
Expanding and rearranging the equation:
\[ y - 2at = -tx + at^3 \]
\[ tx + y = at^3 + 2at \]
Factoring out \( a \):
\[ tx + y = a(t^3 + 2t) \]

### Final Equation
Thus, the equation of the normal to the curve at the point corresponding to the parameter \( t \) is:
\[ tx + y = a(t^3 + 2t) \]

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