The equation of the tangent at the point P(t) ,wheret is any parameter, to the parabola `y^2 = 4ax` is

Options

A. `yt = x+ at^2`
B. `y=xt+at^2`
C. `y=tx`
D. `y=ta//t`

Answer

To find the equation of the tangent at the point \( P(t) \) on the parabola \( y^2 = 4ax \), we can follow these steps:

### Step 1: Identify the coordinates of the point \( P(t) \)
The point \( P(t) \) on the parabola \( y^2 = 4ax \) can be expressed in terms of the parameter \( t \):
- The coordinates are \( P(t) = (at^2, 2at) \).

### Step 2: Find the slope of the tangent line
To find the slope of the tangent line at the point \( P(t) \), we need to differentiate the equation of the parabola with respect to \( x \):
- The equation of the parabola is \( y^2 = 4ax \).
- Differentiating both sides with respect to \( x \), we get:
\[ 2y \frac{dy}{dx} = 4a \]
- Therefore, the slope \( m \) of the tangent line is:
\[ \frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y} \]
- At the point \( P(t) \), where \( y = 2at \):
\[ m = \frac{2a}{2at} = \frac{1}{t} \]

### Step 3: Use the point-slope form of the equation of the tangent
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (at^2, 2at) \) and \( m = \frac{1}{t} \):
\[ y - 2at = \frac{1}{t}(x - at^2) \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation:
1. Multiply both sides by \( t \) to eliminate the fraction:
\[ t(y - 2at) = x - at^2 \]
2. Distributing \( t \):
\[ ty - 2at^2 = x - at^2 \]
3. Rearranging gives:
\[ ty = x + at^2 \]

### Final Equation of the Tangent
Thus, the equation of the tangent at the point \( P(t) \) on the parabola \( y^2 = 4ax \) is:
\[ ty = x + at^2 \]