The coordinate of the point on the curve `x^(2)=4y` which is atleast distance from the line y=x-4 is

Options

A. (a)(2,1)
B. (b)(-2,1)
C. (c)(-2,-1)
D. (d)None of these

Answer

To find the coordinates of the point on the curve \( x^2 = 4y \) that is at least distance from the line \( y = x - 4 \), we can follow these steps:

### Step 1: Understand the Problem
We need to find a point \( (h, k) \) on the curve \( x^2 = 4y \) that minimizes the distance to the line \( y = x - 4 \).

### Step 2: Distance Formula
The distance \( D \) from a point \( (h, k) \) to the line \( Ax + By + C = 0 \) can be calculated using the formula:
\[ D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( y = x - 4 \), we can rewrite it as \( x - y - 4 = 0 \). Here, \( A = 1, B = -1, C = -4 \).

### Step 3: Substitute the Curve Equation
The coordinates \( (h, k) \) must satisfy the curve equation \( x^2 = 4y \). Therefore, we can express \( k \) in terms of \( h \):
\[ k = \frac{h^2}{4} \]

### Step 4: Substitute into the Distance Formula
Now, substituting \( k \) into the distance formula:
\[ D = \frac{|h - \frac{h^2}{4} - 4|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - \frac{h^2}{4} - 4|}{\sqrt{2}} \]

### Step 5: Minimize the Distance
To minimize the distance, we need to minimize the expression \( |h - \frac{h^2}{4} - 4| \). We can drop the absolute value by considering the critical points of the function:
\[ f(h) = h - \frac{h^2}{4} - 4 \]

### Step 6: Differentiate and Find Critical Points
Differentiate \( f(h) \):
\[ f'(h) = 1 - \frac{h}{2} \]
Setting \( f'(h) = 0 \):
\[ 1 - \frac{h}{2} = 0 \implies h = 2 \]

### Step 7: Find Corresponding \( k \)
Now substitute \( h = 2 \) back into the curve equation to find \( k \):
\[ k = \frac{2^2}{4} = \frac{4}{4} = 1 \]

### Step 8: Final Coordinates
Thus, the coordinates of the point on the curve that is at least distance from the line are:
\[ (h, k) = (2, 1) \]

### Conclusion
The required coordinates are \( (2, 1) \).

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