If P is a point on the parabola `y = x^2+ 4` which is closest to the straight line y = 4x – 1, then the co-ordinates of P are :

Options

A. (3, 13)
B. (1,5)
C. (-2, 8)
D. (2, 8)

Answer

To find the coordinates of the point \( P \) on the parabola \( y = x^2 + 4 \) that is closest to the line \( y = 4x - 1 \), we can follow these steps:

### Step 1: Define the point on the parabola
Let the point \( P \) on the parabola be represented as \( (h, k) \). Since \( P \) lies on the parabola, we can express \( k \) in terms of \( h \):
\[ k = h^2 + 4 \]

### Step 2: Write the equation of the line
The equation of the line is given as \( y = 4x - 1 \). We can rearrange this into the standard form:
\[ y - 4x + 1 = 0 \]

### Step 3: Use the distance formula
The distance \( D \) from the point \( P(h, k) \) to the line \( Ax + By + C = 0 \) is given by:
\[ D = \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} \]
For our line \( 4x - y - 1 = 0 \), we have \( A = 4 \), \( B = -1 \), and \( C = 1 \). Thus, the distance becomes:
\[ D = \frac{|4h - k + 1|}{\sqrt{4^2 + (-1)^2}} = \frac{|4h - k + 1|}{\sqrt{16 + 1}} = \frac{|4h - k + 1|}{\sqrt{17}} \]

### Step 4: Substitute \( k \) in the distance formula
Substituting \( k = h^2 + 4 \) into the distance formula gives:
\[ D = \frac{|4h - (h^2 + 4) + 1|}{\sqrt{17}} = \frac{|4h - h^2 - 4 + 1|}{\sqrt{17}} = \frac{|4h - h^2 - 3|}{\sqrt{17}} \]

### Step 5: Minimize the distance
To find the point \( P \) that minimizes the distance \( D \), we need to minimize the expression inside the absolute value:
\[ f(h) = 4h - h^2 - 3 \]
To find the critical points, we differentiate \( f(h) \):
\[ f'(h) = 4 - 2h \]
Setting the derivative equal to zero gives:
\[ 4 - 2h = 0 \implies h = 2 \]

### Step 6: Find \( k \)
Now we substitute \( h = 2 \) back into the equation for \( k \):
\[ k = h^2 + 4 = 2^2 + 4 = 4 + 4 = 8 \]

### Step 7: State the coordinates of point \( P \)
Thus, the coordinates of the point \( P \) that is closest to the line are:
\[ P(2, 8) \]

### Final Answer
The coordinates of \( P \) are \( (2, 8) \).

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