If two tangents drawn from a point P on the parabola `y^2 = 4x` are at right angles, then the locus of P is

Options

A. `2x-1=0`
B. `x=1`
C. `2x+1=0`
D. `x=-1`

Answer

To find the locus of the point \( P \) from which two tangents drawn to the parabola \( y^2 = 4x \) are at right angles, we can follow these steps:

### Step 1: Understand the Parabola
The given parabola is \( y^2 = 4x \). This is a standard form of a parabola that opens to the right.

### Step 2: Equation of Tangents
For a point \( P(x_1, y_1) \) on the parabola, the equation of the tangents can be derived using the formula for the tangent to the parabola \( y^2 = 4ax \):
\[ yy_1 = 2a(x + x_1) \]
Here, \( a = 1 \) (since \( 4a = 4 \)), so the equation of the tangent becomes:
\[ yy_1 = 2(x + x_1) \]

### Step 3: Condition for Perpendicular Tangents
If two tangents are perpendicular, the product of their slopes must equal \(-1\). The slopes of the tangents can be derived from the tangent equation.

The slope of the tangent line can be expressed as:
\[ m = \frac{y}{x + 1} \]
For two tangents from point \( P \) to be perpendicular, we can use the condition:
\[ m_1 \cdot m_2 = -1 \]

### Step 4: Find the Locus
To find the locus of point \( P \), we need to find the relationship between \( x_1 \) and \( y_1 \). Since the tangents are at right angles, we can use the property of the directrix of the parabola.

The equation of the directrix of the parabola \( y^2 = 4x \) is given by:
\[ x = -1 \]
This means that for any point \( P \) from which tangents are drawn at right angles, the \( x \)-coordinate must be \(-1\).

### Step 5: Conclusion
Thus, the locus of point \( P \) is:
\[ x = -1 \]

### Final Answer
The locus of \( P \) is the line:
\[ \boxed{x = -1} \]