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: Understand the curve and the line
The curve given is \( y^2 = 4(x - 10) \), which can be rewritten as:
\[ x = \frac{y^2}{4} + 10 \]
The line can be expressed in slope-intercept form as:
\[ y = -x + 4 \]

### Step 2: Set up the distance formula
We want to find the distance from a point \( (h, k) \) on the curve to the line \( x + y - 4 = 0 \). The distance \( d \) from a point \( (h, k) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} \]
For our line, \( A = 1, B = 1, C = -4 \). Thus, the distance becomes:
\[ d = \frac{|h + k - 4|}{\sqrt{1^2 + 1^2}} = \frac{|h + k - 4|}{\sqrt{2}} \]

### Step 3: Substitute the curve equation
Since \( (h, k) \) lies on the curve, we have:
\[ k^2 = 4(h - 10) \quad \Rightarrow \quad h = \frac{k^2}{4} + 10 \]
Substituting \( h \) into the distance formula gives:
\[ d = \frac{\left|\frac{k^2}{4} + 10 + k - 4\right|}{\sqrt{2}} = \frac{\left|\frac{k^2}{4} + k + 6\right|}{\sqrt{2}} \]

### Step 4: Minimize the distance
To minimize the distance, we can minimize the expression \( \left|\frac{k^2}{4} + k + 6\right| \). We can ignore the absolute value for minimization purposes and focus on:
\[ f(k) = \frac{k^2}{4} + k + 6 \]
To find the minimum, we differentiate \( f(k) \):
\[ f'(k) = \frac{1}{2}k + 1 \]
Setting the derivative to zero:
\[ \frac{1}{2}k + 1 = 0 \quad \Rightarrow \quad k = -2 \]

### Step 5: Find corresponding \( h \)
Now substituting \( k = -2 \) back into the equation for \( h \):
\[ h = \frac{(-2)^2}{4} + 10 = \frac{4}{4} + 10 = 1 + 10 = 11 \]

### Step 6: Conclusion
Thus, the point on the curve that is nearest to the line is:
\[ (h, k) = (11, -2) \]

### Final Answer
The point on the curve \( y^2 = 4(x - 10) \) which is nearest to the line \( x + y = 4 \) is \( (11, -2) \).

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