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

Options

A. (2,1)
B. (-2,1)
C. (-2,-1)
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 given curve and line
The curve \( x^2 = 4y \) is a parabola that opens upwards. The line \( y = x - 4 \) is a straight line with a slope of 1.

**Hint:** Sketch the parabola and the line to visualize their positions relative to each other.

### Step 2: Find the slope of the line
The slope of the line \( y = x - 4 \) is 1. We need to find a point on the parabola where the tangent line has the same slope (1).

**Hint:** Recall that the slope of the tangent line to the curve can be found by differentiating the equation of the curve.

### Step 3: Differentiate the curve
Differentiate the equation \( x^2 = 4y \) with respect to \( x \):
\[ \frac{d}{dx}(x^2) = \frac{d}{dx}(4y) \]
This gives:
\[ 2x = 4 \frac{dy}{dx} \]
So,
\[ \frac{dy}{dx} = \frac{2x}{4} = \frac{x}{2} \]

**Hint:** Set the derivative equal to the slope of the line to find the x-coordinate of the point.

### Step 4: Set the derivative equal to the slope
We want the slope of the tangent to equal the slope of the line:
\[ \frac{x}{2} = 1 \]
Solving for \( x \):
\[ x = 2 \]

**Hint:** Now that you have \( x \), substitute it back into the equation of the parabola to find \( y \).

### Step 5: Substitute \( x \) back into the parabola
Substituting \( x = 2 \) into the parabola equation \( x^2 = 4y \):
\[ 2^2 = 4y \]
This simplifies to:
\[ 4 = 4y \implies y = 1 \]

**Hint:** The coordinates of the point on the parabola are now found.

### Step 6: Write the final answer
The coordinates of the point on the curve \( x^2 = 4y \) that is at least distance from the line \( y = x - 4 \) are \( (2, 1) \).

**Final Answer:** The point is \( (2, 1) \).