Find the equation of tangent of the curve `x=at^(2), y = 2at` at point `t`.

Options

A. `x-ty + at^(3) = 0`
B. `x-y + at^(2) = 0`
C. `x-ty + at^(2) = 0`
D. `tx-y + at^(2) = 0`

Explanation

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

### Step 1: Find the coordinates of the point on the curve at parameter \( t \).

The coordinates of the point on the curve can be found by substituting \( t \) into the equations for \( x \) and \( y \):
\[ x_1 = at^2 \]
\[ y_1 = 2at \]

### Step 2: Find the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \).

Differentiate \( x \) and \( y \) with respect to \( t \):
\[ \frac{dx}{dt} = \frac{d}{dt}(at^2) = 2at \]
\[ \frac{dy}{dt} = \frac{d}{dt}(2at) = 2a \]

### Step 3: Find the slope \( m \) of the tangent line.

The slope of the tangent line can be found using the formula:
\[ m = \frac{dy/dt}{dx/dt} = \frac{2a}{2at} = \frac{1}{t} \]

### Step 4: Write the equation of the tangent line.

The equation of the tangent line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m \), \( x_1 \), and \( y_1 \):
\[ y - 2at = \frac{1}{t}(x - at^2) \]

### Step 5: Simplify the equation.

Multiply both sides by \( t \) to eliminate the fraction:
\[ t(y - 2at) = x - at^2 \]
Expanding and rearranging gives:
\[ ty - 2at^2 = x - at^2 \]
\[ ty - x + at^2 = 2at^2 \]
\[ ty - x = 2at^2 - at^2 \]
\[ ty - x = at^2 \]

### Final Equation of the Tangent Line:

Thus, the equation of the tangent line at the point corresponding to parameter \( t \) is:
\[ ty - x = at^2 \]

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