The locus of the point of intersection of perpendicular tangents to the parabola `y^(2)=4ax` is

Options

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

Answer

To find the locus of the point of intersection of perpendicular tangents to the parabola \( y^2 = 4ax \), we can follow these steps:

### Step 1: Write the equation of the parabola
The given parabola is \( y^2 = 4ax \).

### Step 2: Write the equation of the tangent to the parabola
The equation of the tangent to the parabola at a point with slope \( m \) is given by:
\[ y = mx - am^2 + a \]
This can be rearranged to:
\[ y = mx - a(m^2 - 1) \]

### Step 3: Find the condition for perpendicular tangents
For two tangents to be perpendicular, the product of their slopes must equal -1:
\[ m_1 m_2 = -1 \]

### Step 4: Use the quadratic formula for tangents
The tangents can be represented in terms of their slopes \( m_1 \) and \( m_2 \). The sum of the slopes \( m_1 + m_2 \) and the product of the slopes \( m_1 m_2 \) can be derived from the quadratic equation formed by the tangents.

From the quadratic equation:
\[ y = mx - a(m^2 - 1) \]
we can express it as:
\[ am^2 - ym + (y - a) = 0 \]
where \( a \) is the coefficient of \( m^2 \), \( -y \) is the coefficient of \( m \), and \( (y - a) \) is the constant term.

### Step 5: Apply Vieta's formulas
Using Vieta's formulas, we have:
- Sum of roots: \( m_1 + m_2 = \frac{y}{a} \)
- Product of roots: \( m_1 m_2 = \frac{y - a}{a} \)

### Step 6: Set up the perpendicular condition
Since \( m_1 m_2 = -1 \), we can set up the equation:
\[ \frac{y - a}{a} = -1 \]
This simplifies to:
\[ y - a = -a \implies y = 0 \]

### Step 7: Find the x-coordinate of the intersection point
Now substituting \( y = 0 \) into the sum of roots equation:
\[ m_1 + m_2 = \frac{0}{a} = 0 \implies m_1 = -m_2 \]
This means that the slopes are indeed perpendicular.

### Step 8: Determine the locus
From the earlier derived equations, we can find the x-coordinate of the point of intersection:
Using the product of roots:
\[ m_1 m_2 = -1 \implies \frac{y}{a} = -1 \implies y = -a \]
Since we have \( y = 0 \), we can express \( x \) in terms of \( a \):
\[ x = -a \]

### Conclusion
The locus of the point of intersection of the perpendicular tangents to the parabola \( y^2 = 4ax \) is:
\[ x + a = 0 \quad \text{or} \quad x = -a \]