Coordinates of the point on the stratight line `x +y =4,` which is nearest to the parabola `y ^(2) =4 (x -10) is (a, b),` then `a ^(2)+b ^(2)` is `"______"`

Answer

To find the coordinates of the point on the straight line \( x + y = 4 \) that is nearest to the parabola \( y^2 = 4(x - 10) \), we will follow these steps:

### Step 1: Understand the Problem
We need to find the point \( (a, b) \) on the line \( x + y = 4 \) that is closest to the parabola \( y^2 = 4(x - 10) \).

### Step 2: Rewrite the Line Equation
From the line equation \( x + y = 4 \), we can express \( y \) in terms of \( x \):
\[ y = 4 - x \]

### Step 3: Substitute into the Parabola Equation
Substituting \( y = 4 - x \) into the parabola equation \( y^2 = 4(x - 10) \):
\[ (4 - x)^2 = 4(x - 10) \]

### Step 4: Expand and Simplify
Expanding the left side:
\[ 16 - 8x + x^2 = 4x - 40 \]
Rearranging gives:
\[ x^2 - 12x + 56 = 0 \]

### Step 5: Solve the Quadratic Equation
We can solve the quadratic equation \( x^2 - 12x + 56 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -12, c = 56 \):
\[ x = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 56}}{2 \cdot 1} \]
\[ x = \frac{12 \pm \sqrt{144 - 224}}{2} \]
\[ x = \frac{12 \pm \sqrt{-80}}{2} \]
Since the discriminant is negative, we need to check for the minimum distance using a different approach.

### Step 6: Find the Distance Function
The distance \( D \) from a point \( (x, y) \) on the line to a point \( (x_0, y_0) \) on the parabola can be expressed as:
\[ D^2 = (x - x_0)^2 + (y - y_0)^2 \]
Substituting \( y = 4 - x \):
\[ D^2 = (x - x_0)^2 + ((4 - x) - y_0)^2 \]

### Step 7: Minimize the Distance
To find the minimum distance, we can use the method of Lagrange multipliers or directly find the point on the line that minimizes the distance to the parabola.

### Step 8: Find the Coordinates
After analyzing the slopes and using the conditions for tangents, we find that the coordinates of the nearest point are:
\[ (a, b) = \left(\frac{17}{2}, -\frac{9}{2}\right) \]

### Step 9: Calculate \( a^2 + b^2 \)
Now we calculate \( a^2 + b^2 \):
\[ a^2 + b^2 = \left(\frac{17}{2}\right)^2 + \left(-\frac{9}{2}\right)^2 \]
\[ = \frac{289}{4} + \frac{81}{4} = \frac{370}{4} = 92.5 \]

Thus, the final answer is:
\[ \boxed{92.5} \]