To find the length of the chord of the parabola \( y^2 = x \) that is bisected at the point \( (2, 1) \), we can follow these steps:
### Step 1: Understand the parabola and the midpoint
The equation of the parabola is given as \( y^2 = x \). The point \( (2, 1) \) is the midpoint of the chord we are trying to find.
### Step 2: Parametrize the points on the parabola
We can represent points on the parabola using a parameter \( t \). For any point on the parabola, we can write:
- \( P_1(t_1) = (t_1^2, t_1) \)
- \( P_2(t_2) = (t_2^2, t_2) \)
### Step 3: Use the midpoint formula
Since \( (2, 1) \) is the midpoint of the chord, we can use the midpoint formula:
\[
\left( \frac{t_1^2 + t_2^2}{2}, \frac{t_1 + t_2}{2} \right) = (2, 1)
\]
From this, we can derive two equations:
1. \( \frac{t_1^2 + t_2^2}{2} = 2 \) → \( t_1^2 + t_2^2 = 4 \) (Equation 1)
2. \( \frac{t_1 + t_2}{2} = 1 \) → \( t_1 + t_2 = 2 \) (Equation 2)
### Step 4: Solve the equations
From Equation 2, we can express \( t_1 \) in terms of \( t_2 \):
\[
t_1 = 2 - t_2
\]
Now, substitute \( t_1 \) into Equation 1:
\[
(2 - t_2)^2 + t_2^2 = 4
\]
Expanding this gives:
\[
4 - 4t_2 + t_2^2 + t_2^2 = 4
\]
This simplifies to:
\[
2t_2^2 - 4t_2 = 0
\]
Factoring out \( 2t_2 \):
\[
2t_2(t_2 - 2) = 0
\]
Thus, we have two solutions:
1. \( t_2 = 0 \)
2. \( t_2 = 2 \)
### Step 5: Find corresponding \( t_1 \) values
Using \( t_1 = 2 - t_2 \):
- If \( t_2 = 0 \), then \( t_1 = 2 \).
- If \( t_2 = 2 \), then \( t_1 = 0 \).
### Step 6: Determine the points on the parabola
Now we can find the coordinates of the points \( P_1 \) and \( P_2 \):
- For \( t_1 = 2 \) and \( t_2 = 0 \):
- \( P_1(2) = (2^2, 2) = (4, 2) \)
- \( P_2(0) = (0^2, 0) = (0, 0) \)
### Step 7: Calculate the length of the chord
The length of the chord between points \( P_1(4, 2) \) and \( P_2(0, 0) \) can be calculated using the distance formula:
\[
\text{Length} = \sqrt{(4 - 0)^2 + (2 - 0)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]
### Final Answer
The length of the chord is \( 2\sqrt{5} \).
---
