Let PQ be a focal chord of the parabola `y^(2)=4x`. If the centre of a circle having PQ as its diameter lies on the line `sqrt5y+4=0`, then length of the chord PQ, is

Options

A. `36/5`
B. `26/5`
C. `(36sqrt5)/5`
D. `(26sqrt5)/5`

Explanation

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the Parabola and Focal Chord
The given parabola is \( y^2 = 4x \). The focal point of this parabola is at \( (1, 0) \). A focal chord is a line segment that passes through the focus and has endpoints on the parabola.

### Step 2: Parameterization of Points on the Parabola
We can parameterize points on the parabola using the parameter \( t \):
- Let \( P \) correspond to parameter \( t_1 \): \( P(t_1) = (t_1^2, 2t_1) \)
- Let \( Q \) correspond to parameter \( t_2 \): \( Q(t_2) = (t_2^2, 2t_2) \)

### Step 3: Condition for Focal Chord
For \( PQ \) to be a focal chord, the product of the parameters must satisfy:
\[ t_1 \cdot t_2 = -1 \]

### Step 4: Midpoint of Chord PQ
The midpoint \( C \) of the chord \( PQ \) is given by:
\[ C = \left( \frac{t_1^2 + t_2^2}{2}, \frac{2t_1 + 2t_2}{2} \right) = \left( \frac{t_1^2 + t_2^2}{2}, t_1 + t_2 \right) \]

### Step 5: Center of Circle Condition
The center \( C \) lies on the line \( \sqrt{5}y + 4 = 0 \). Therefore, substituting the y-coordinate of \( C \):
\[ \sqrt{5}(t_1 + t_2) + 4 = 0 \quad \Rightarrow \quad t_1 + t_2 = -\frac{4}{\sqrt{5}} \]

### Step 6: Finding Length of Chord PQ
The length \( L \) of the chord \( PQ \) can be calculated using the distance formula:
\[ L = \sqrt{(t_1^2 - t_2^2)^2 + (2t_1 - 2t_2)^2} \]
This simplifies to:
\[ L = \sqrt{(t_1 - t_2)^2(t_1 + t_2)^2 + 4(t_1 - t_2)^2} = |t_1 - t_2| \sqrt{(t_1 + t_2)^2 + 4} \]

### Step 7: Substitute Known Values
We know:
- \( t_1 + t_2 = -\frac{4}{\sqrt{5}} \)
- \( t_1 t_2 = -1 \)

Using the identity \( t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2t_1t_2 \):
\[ t_1^2 + t_2^2 = \left(-\frac{4}{\sqrt{5}}\right)^2 - 2(-1) = \frac{16}{5} + 2 = \frac{16}{5} + \frac{10}{5} = \frac{26}{5} \]

### Step 8: Calculate Length of Chord
Now substituting back into the length formula:
\[ L = |t_1 - t_2| \sqrt{\left(-\frac{4}{\sqrt{5}}\right)^2 + 4} \]
We need \( |t_1 - t_2| \):
\[ |t_1 - t_2| = \sqrt{(t_1 + t_2)^2 - 4t_1t_2} = \sqrt{\left(-\frac{4}{\sqrt{5}}\right)^2 - 4(-1)} = \sqrt{\frac{16}{5} + 4} = \sqrt{\frac{16}{5} + \frac{20}{5}} = \sqrt{\frac{36}{5}} = \frac{6}{\sqrt{5}} \]

Finally, substituting everything into the length formula:
\[ L = \frac{6}{\sqrt{5}} \sqrt{\frac{16}{5} + 4} = \frac{6}{\sqrt{5}} \sqrt{\frac{36}{5}} = \frac{6}{\sqrt{5}} \cdot \frac{6}{\sqrt{5}} = \frac{36}{5} \]

### Final Answer
Thus, the length of the chord \( PQ \) is \( \frac{36}{5} \).

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