The shortest distance between the lines `y-x=1` and the curve `x=y^2` is

Options

A. `(3sqrt(2))/(8)`
B. `(2sqrt(2))/(8)`
C. `(3sqrt(2))/(5)`
D. `(sqrt(3))/(4)`

Answer

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

### Step 1: Rewrite the equations
First, let's rewrite the equations in a more useful form:
- The line can be expressed as \( y = x + 1 \).
- The curve is already in a suitable form: \( x = y^2 \).

### Step 2: Substitute the line equation into the curve equation
We will substitute \( y = x + 1 \) into the curve equation \( x = y^2 \):
\[ x = (x + 1)^2 \]
Expanding this gives:
\[ x = x^2 + 2x + 1 \]
Rearranging the equation:
\[ 0 = x^2 + x + 1 \]

### Step 3: Solve the quadratic equation
Now we need to solve the quadratic equation \( x^2 + x + 1 = 0 \). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 1, c = 1 \):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} \]
Since the discriminant is negative, there are no real solutions, which means the line does not intersect the curve.

### Step 4: Find the distance between the line and the curve
To find the shortest distance, we will use the distance formula between a point on the curve and the line. We can express the distance \( D \) between a point \( (y^2, y) \) on the curve and the line \( y = x + 1 \):
\[ D = \frac{|y - (y^2 + 1)|}{\sqrt{1^2 + (-1)^2}} = \frac{|y - y^2 - 1|}{\sqrt{2}} \]
This simplifies to:
\[ D = \frac{|y^2 - y + 1|}{\sqrt{2}} \]

### Step 5: Minimize the distance
To minimize \( D \), we can minimize the expression \( |y^2 - y + 1| \). The function \( f(y) = y^2 - y + 1 \) is a quadratic function that opens upwards. The vertex of this parabola occurs at:
\[ y = -\frac{b}{2a} = -\frac{-1}{2 \cdot 1} = \frac{1}{2} \]
Calculating \( f\left(\frac{1}{2}\right) \):
\[ f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \frac{1}{2} + 1 = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{3}{4} \]

### Step 6: Calculate the minimum distance
Thus, the minimum distance \( D \) is:
\[ D = \frac{|\frac{3}{4}|}{\sqrt{2}} = \frac{3}{4\sqrt{2}} = \frac{3\sqrt{2}}{8} \]

### Final Answer
The shortest distance between the line \( y - x = 1 \) and the curve \( x = y^2 \) is \( \frac{3\sqrt{2}}{8} \).