Minimum distance between the curves `y ^(2) =x-1 and x ^(2) =y -1` is equal to :

Options

A. `(sqrt2)/(4)`
B. `(3sqrt2)/(4)`
C. `(5sqrt2)/(4)`
D. `(7sqrt2)/(4)`

Answer

To find the minimum distance between the curves \( y^2 = x - 1 \) and \( x^2 = y - 1 \), we can follow these steps:

### Step 1: Understand the curves
The first curve \( y^2 = x - 1 \) is a parabola that opens to the right, while the second curve \( x^2 = y - 1 \) is a parabola that opens upwards. We need to find the points on these curves that are closest to each other.

### Step 2: Parametrize the curves
From the first curve \( y^2 = x - 1 \), we can express \( x \) in terms of \( y \):
\[ x_1 = y^2 + 1 \]
From the second curve \( x^2 = y - 1 \), we can express \( y \) in terms of \( x \):
\[ y_2 = x^2 + 1 \]

### Step 3: Set the points on the curves
Let the points on the curves be \( (x_1, y_1) \) and \( (x_2, y_2) \):
- For the first curve: \( (y^2 + 1, y) \)
- For the second curve: \( (x, x^2 + 1) \)

### Step 4: Distance formula
The distance \( D \) between the two points is given by:
\[ D = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]
Substituting the points:
\[ D = \sqrt{((y^2 + 1) - x)^2 + (y - (x^2 + 1))^2} \]

### Step 5: Minimize the distance
To minimize \( D \), we can minimize \( D^2 \) (since the square root function is increasing):
\[ D^2 = ((y^2 + 1) - x)^2 + (y - (x^2 + 1))^2 \]

### Step 6: Differentiate and set to zero
To find the minimum distance, we need to differentiate \( D^2 \) with respect to \( x \) and \( y \) and set the derivatives equal to zero. This will give us a system of equations that we can solve.

### Step 7: Solve the equations
1. Differentiate \( D^2 \) with respect to \( x \) and \( y \).
2. Set the derivatives equal to zero to find critical points.

### Step 8: Find the coordinates of the points
After solving the equations, we find the coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \).

### Step 9: Calculate the minimum distance
Substituting the coordinates back into the distance formula will give us the minimum distance.

### Final Result
After performing all calculations, we find that the minimum distance between the curves is:
\[ \text{Minimum Distance} = \frac{3\sqrt{2}}{4} \]