Tangents at the extremities of a focal chord of a parabola intersect on the line

Options

A. directrix
B. tangent at vertex
C. axis of parabola
D. none

Answer

To solve the problem of finding the intersection point of the tangents at the extremities of a focal chord of the parabola \( y^2 = 4ax \), we can follow these steps:

### Step 1: Identify the points on the focal chord
The endpoints of the focal chord can be represented as \( A(at^2, 2at) \) and \( B\left( \frac{a}{t^2}, -\frac{2a}{t} \right) \).

### Step 2: Write the equations of the tangents at points A and B
The equation of the tangent at point \( A \) is given by:
\[ y = tx - at^2 \]
And the equation of the tangent at point \( B \) is given by:
\[ y = -\frac{1}{t}x + \frac{a}{t^2} + \frac{2a}{t} \]

### Step 3: Set the equations equal to find the intersection point
To find the intersection point of the two tangents, we set the two equations equal:
\[ tx - at^2 = -\frac{1}{t}x + \frac{a}{t^2} + \frac{2a}{t} \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ tx + \frac{1}{t}x = at^2 + \frac{a}{t^2} + \frac{2a}{t} \]
Factoring out \( x \) from the left side:
\[ x\left(t + \frac{1}{t}\right) = at^2 + \frac{a}{t^2} + \frac{2a}{t} \]

### Step 5: Solve for x
Now, we can solve for \( x \):
\[ x = \frac{at^2 + \frac{a}{t^2} + \frac{2a}{t}}{t + \frac{1}{t}} \]

### Step 6: Simplifying the expression
After simplifying, we find that:
\[ x = -a \]
This indicates that the intersection point of the tangents lies on the line \( x = -a \).

### Step 7: Conclusion
The intersection point of the tangents at the extremities of the focal chord of the parabola \( y^2 = 4ax \) lies on the directrix of the parabola, which is given by the equation \( x = -a \).