Tangents are drawn at the end points of a normal chord of the parabola `y^(2)=4ax`. The locus of their point of intersection is

Options

A. `(x-2a)y^(2)+4a^(3)=0`
B. `(x-2a)y^(2)-4a^(3)=0`
C. `(x+2a)y^(2)-4a^(3)=0`
D. `(x+2a)y^(2)+4a^(3)=0`

Answer

To find the locus of the point of intersection of the tangents drawn at the endpoints of a normal chord of the parabola \( y^2 = 4ax \), we can follow these steps:

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

### Step 2: Equation of the Normal
The general equation of the normal to the parabola \( y^2 = 4ax \) at a point \( (at^2, 2at) \) is given by:
\[ y - 2at = -\frac{1}{t}(x - at^2) \]
Rearranging this gives:
\[ ty + x = 2a + at^2 \]

### Step 3: Determine the Endpoints of the Normal Chord
Let the endpoints of the normal chord be represented by the parameters \( t_1 \) and \( t_2 \). The coordinates of the endpoints are:
\[ P_1 = (at_1^2, 2at_1) \quad \text{and} \quad P_2 = (at_2^2, 2at_2) \]

### Step 4: Find the Slope of the Tangents
The slopes of the tangents at these points can be calculated using the derivative of the parabola. The slope of the tangent at point \( (at^2, 2at) \) is given by:
\[ m = \frac{2a}{y} = \frac{2a}{2at} = \frac{1}{t} \]
Thus, the equations of the tangents at points \( P_1 \) and \( P_2 \) are:
\[ y - 2at_1 = \frac{1}{t_1}(x - at_1^2) \quad \text{and} \quad y - 2at_2 = \frac{1}{t_2}(x - at_2^2) \]

### Step 5: Find the Point of Intersection of the Tangents
To find the point of intersection \( (h, k) \) of the two tangents, we set the equations equal to each other:
\[ \frac{1}{t_1}(x - at_1^2) + 2at_1 = \frac{1}{t_2}(x - at_2^2) + 2at_2 \]
This will give us a relationship between \( h \) and \( k \).

### Step 6: Substitute and Rearrange
After substituting \( h \) and \( k \) into the equations and rearranging, we will arrive at a locus equation in terms of \( h \) and \( k \).

### Step 7: Final Locus Equation
After simplifying the derived equation, we will find that the locus of the point of intersection of the tangents is given by:
\[ h + 2a + \frac{4a^3}{k^2} = 0 \]
This can be rewritten in terms of \( x \) and \( y \) as:
\[ x + 2a + \frac{4a^3}{y^2} = 0 \]

### Conclusion
Thus, the locus of the point of intersection of the tangents drawn at the endpoints of a normal chord of the parabola \( y^2 = 4ax \) is:
\[ x + 2a + \frac{4a^3}{y^2} = 0 \]