The length of the shortest path that begins at the point `(-1, 1)`, touches the x-axis and then ends at a point on the parabola `(x-y)^(2)=2(x+y-4)`, is :

Options

A. `3sqrt(2)`
B. 5
C. `4sqrt(10)`
D. 13

Answer

To find the length of the shortest path that begins at the point \((-1, 1)\), touches the x-axis, and then ends at a point on the parabola \((x - y)^2 = 2(x + y - 4)\), we can follow these steps:

### Step 1: Understand the Problem
We need to find a point on the x-axis (let's call it \(A\)) where the path touches, and then find a point on the parabola (let's call it \(B\)). The total length of the path will be the sum of the lengths from \((-1, 1)\) to \(A\) and from \(A\) to \(B\).

### Step 2: Identify the Coordinates of Point A
Let the coordinates of point \(A\) on the x-axis be \((x_A, 0)\). The y-coordinate is 0 because it lies on the x-axis.

### Step 3: Calculate the Length from Point O to Point A
Using the distance formula, the distance from point \(O(-1, 1)\) to point \(A(x_A, 0)\) is:
\[ OA = \sqrt{(x_A - (-1))^2 + (0 - 1)^2} = \sqrt{(x_A + 1)^2 + 1} \]

### Step 4: Identify Point B on the Parabola
The parabola is given by the equation \((x - y)^2 = 2(x + y - 4)\). We can rearrange this to find points on the parabola.

### Step 5: Express Point B in Terms of x
Let point \(B\) on the parabola be \((x_B, y_B)\). We can use the parabola equation to express \(y_B\) in terms of \(x_B\):
\[ (x_B - y_B)^2 = 2(x_B + y_B - 4) \]

### Step 6: Calculate the Length from Point A to Point B
Using the distance formula again, the distance from point \(A(x_A, 0)\) to point \(B(x_B, y_B)\) is:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - 0)^2} = \sqrt{(x_B - x_A)^2 + y_B^2} \]

### Step 7: Total Length of the Path
The total length \(L\) of the path is:
\[ L = OA + AB = \sqrt{(x_A + 1)^2 + 1} + \sqrt{(x_B - x_A)^2 + y_B^2} \]

### Step 8: Minimize the Length
To find the shortest path, we need to minimize \(L\) with respect to \(x_A\) and \(x_B\). This typically involves calculus, where we take the derivative of \(L\) and set it to zero to find critical points.

### Step 9: Solve for Specific Points
After finding the critical points, we can evaluate the length \(L\) at those points to determine the minimum length.

### Step 10: Conclusion
Finally, we conclude with the minimum length of the path.

---