locus of point of reflection of point `(a,0)` w.r.t. the line `yt=x+at^2` is given by:

Options

A. ` x - a = 0 `
B. ` y - a = 0`
C. `x + a = 0 `
D. `y + a = 0 `

Answer

To find the locus of the point of reflection of the point \( (a, 0) \) with respect to the line \( yt = x + at^2 \), we can follow these steps:

### Step 1: Understand the given line
The line \( yt = x + at^2 \) can be rewritten in the standard form of a line. Here, \( t \) is a parameter. This line represents a tangent to the parabola \( y^2 = 4ax \) at some point \( P(t) \).

### Step 2: Identify the focus and directrix of the parabola
The parabola \( y^2 = 4ax \) has its focus at \( (a, 0) \) and its directrix is given by the equation \( x = -a \).

### Step 3: Use the property of reflection
The property of reflection states that the image of the focus (or any point) with respect to a tangent line lies on the directrix of the parabola. Therefore, the reflection of the point \( (a, 0) \) will lie on the directrix \( x = -a \).

### Step 4: Set up the locus equation
Since the locus of the point of reflection must satisfy the equation of the directrix, we can express this as:
\[ x + a = 0 \]
which simplifies to:
\[ x = -a \]

### Step 5: Conclusion
Thus, the locus of the point of reflection of the point \( (a, 0) \) with respect to the line \( yt = x + at^2 \) is given by the equation of the directrix:
\[ x + a = 0 \]

### Final Answer
The locus is \( x + a = 0 \).

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