Point on the curve `y ^(2) = 4 (x -10) ` which is nearest to the line `x + y =4` may be

Options

A. `(11,2)`
B. `(10,0)`
C. `(11,-2)`
D. none of these

Answer

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

### Step 1: Rewrite the equations
The equation of the curve can be rewritten as:
\[ y^2 = 4x - 40 \]
The equation of the line can be rewritten as:
\[ y = 4 - x \]

### Step 2: Find the distance from a point on the curve to the line
Let a point on the curve be \( (x, y) \). The distance \( d \) from the point \( (x, y) \) to the line \( x + y - 4 = 0 \) can be calculated using the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where \( A = 1, B = 1, C = -4 \), and \( (x_1, y_1) = (x, y) \).

Substituting the values:
\[ d = \frac{|1 \cdot x + 1 \cdot y - 4|}{\sqrt{1^2 + 1^2}} = \frac{|x + y - 4|}{\sqrt{2}} \]

### Step 3: Substitute \( y \) from the curve equation
From the curve equation \( y^2 = 4(x - 10) \), we can express \( y \) in terms of \( x \):
\[ y = \pm 2\sqrt{x - 10} \]
We can substitute \( y \) into the distance formula:
\[ d = \frac{|x + 2\sqrt{x - 10} - 4|}{\sqrt{2}} \quad \text{(for the positive root)} \]
or
\[ d = \frac{|x - 2\sqrt{x - 10} - 4|}{\sqrt{2}} \quad \text{(for the negative root)} \]

### Step 4: Minimize the distance
To find the point on the curve that is nearest to the line, we need to minimize the distance \( d \). We can minimize \( |x + 2\sqrt{x - 10} - 4| \) or \( |x - 2\sqrt{x - 10} - 4| \).

Let’s minimize \( f(x) = x + 2\sqrt{x - 10} - 4 \).

### Step 5: Differentiate and find critical points
To minimize \( f(x) \), we differentiate it:
\[ f'(x) = 1 + \frac{2}{2\sqrt{x - 10}} \cdot \frac{1}{2} = 1 + \frac{1}{\sqrt{x - 10}} \]
Setting \( f'(x) = 0 \):
\[ 1 + \frac{1}{\sqrt{x - 10}} = 0 \]
This equation has no solution since \( \sqrt{x - 10} \) cannot be negative.

### Step 6: Check endpoints and behavior
Since \( f'(x) \) is always positive for \( x > 10 \), \( f(x) \) is increasing in the interval \( x > 10 \). We can check the minimum distance at \( x = 10 \):
\[ y = 2\sqrt{10 - 10} = 0 \]
So the point is \( (10, 0) \).

### Step 7: Calculate the distance from \( (10, 0) \) to the line
Now we calculate the distance from \( (10, 0) \) to the line \( x + y = 4 \):
\[ d = \frac{|10 + 0 - 4|}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2} \]

### Step 8: Check other point on the curve
For the negative root \( y = -2\sqrt{x - 10} \):
\[ f(x) = x - 2\sqrt{x - 10} - 4 \]
Following similar steps, we can find that the minimum distance occurs at the same point.

### Conclusion
The point on the curve \( y^2 = 4(x - 10) \) which is nearest to the line \( x + y = 4 \) is:
\[ \boxed{(11, -2)} \]