Let PQ be a focal chord of the parabola `y^2 = 4ax` The tangents to the parabola at P and Q meet at a point lying on the line `y = 2x + a, a > 0`. Length of chord PQ is

Options

A. 7a
B. 5a
C. 2a
D. 3a

Explanation

To find the length of the focal chord \( PQ \) of the parabola \( y^2 = 4ax \), we will follow these steps:

### Step 1: Parametric Representation of Points P and Q
The points \( P \) and \( Q \) on the parabola can be represented in parametric form as:
- \( P(at_1^2, 2at_1) \)
- \( Q(at_2^2, 2at_2) \)

### Step 2: Equation of Tangents at Points P and Q
The equations of the tangents at points \( P \) and \( Q \) are given by:
- For point \( P \): \( y = t_1x + at_1^2 \)
- For point \( Q \): \( y = t_2x + at_2^2 \)

### Step 3: Intersection Point of Tangents
Let the intersection point of the tangents at \( P \) and \( Q \) be \( R \). The coordinates of \( R \) can be found by solving the equations of the tangents simultaneously.

### Step 4: Condition for Intersection Point R
Given that point \( R \) lies on the line \( y = 2x + a \), we can substitute the coordinates of \( R \) into this equation. The coordinates of \( R \) can be expressed as:
- \( R(at_1t_2, a(t_1 + t_2)) \)

### Step 5: Setting Up the Equation
Using the condition that \( R \) lies on the line \( y = 2x + a \):
\[ a(t_1 + t_2) = 2(at_1t_2) + a \]
Simplifying this gives:
\[ t_1 + t_2 = 2t_1t_2 + 1 \]

### Step 6: Using Focal Chord Condition
For a focal chord, we know that:
\[ t_1t_2 = -1 \]
Substituting \( t_1t_2 = -1 \) into the equation from Step 5 gives:
\[ t_1 + t_2 = 2(-1) + 1 = -1 \]

### Step 7: Finding \( t_1 - t_2 \)
Now we have:
- \( t_1 + t_2 = -1 \)
- \( t_1t_2 = -1 \)

Using the identity \( (t_1 - t_2)^2 = (t_1 + t_2)^2 - 4t_1t_2 \):
\[ (t_1 - t_2)^2 = (-1)^2 - 4(-1) = 1 + 4 = 5 \]
Thus, \( t_1 - t_2 = \sqrt{5} \).

### Step 8: Length of Chord PQ
The length of chord \( PQ \) is given by:
\[ PQ = \sqrt{(at_1^2 - at_2^2)^2 + (2at_1 - 2at_2)^2} \]
This can be simplified as:
\[ PQ = \sqrt{a^2(t_1^2 - t_2^2)^2 + 4a^2(t_1 - t_2)^2} \]
Using the difference of squares:
\[ t_1^2 - t_2^2 = (t_1 + t_2)(t_1 - t_2) = (-1)(\sqrt{5}) = -\sqrt{5} \]
Thus:
\[ PQ = \sqrt{a^2(-\sqrt{5})^2 + 4a^2(\sqrt{5})^2} \]
Calculating this gives:
\[ PQ = \sqrt{5a^2 + 20a^2} = \sqrt{25a^2} = 5a \]

### Final Answer
The length of the chord \( PQ \) is \( 5a \).

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