If `P(h,k)` be a point on the parabola `x=4y^(2)`,which is nearest to the point `Q(0,33)`,then the distance of `P` from the directrix of the parabola `y^(2)=4(x+y)` is equal to :

Options

A. `4`
B. `2`
C. `8`
D. `6`

Answer

To solve the problem, we need to find the point \( P(h, k) \) on the parabola \( x = 4y^2 \) that is nearest to the point \( Q(0, 33) \), and then calculate the distance from this point \( P \) to the directrix of the parabola \( y^2 = 4(x+y) \).

### Step 1: Identify the parabola and its properties
The given parabola is \( x = 4y^2 \). This can be rewritten in standard form as \( y^2 = \frac{x}{4} \). The vertex of this parabola is at the origin (0, 0), and it opens to the right.

### Step 2: Find the coordinates of point \( P \)
Let \( P(h, k) \) be a point on the parabola. Since \( P \) lies on the parabola, we have:
\[ h = 4k^2 \]
We want to minimize the distance from \( P \) to \( Q(0, 33) \). The distance \( D \) can be expressed as:
\[ D = \sqrt{(h - 0)^2 + (k - 33)^2} = \sqrt{h^2 + (k - 33)^2} \]
Substituting \( h = 4k^2 \) into the distance formula gives:
\[ D = \sqrt{(4k^2)^2 + (k - 33)^2} = \sqrt{16k^4 + (k - 33)^2} \]

### Step 3: Minimize the distance
To minimize \( D \), we can minimize \( D^2 \):
\[ D^2 = 16k^4 + (k - 33)^2 \]
Expanding the second term:
\[ D^2 = 16k^4 + (k^2 - 66k + 1089) \]
Combining like terms:
\[ D^2 = 16k^4 + k^2 - 66k + 1089 \]

### Step 4: Differentiate and find critical points
Now, we differentiate \( D^2 \) with respect to \( k \) and set it to zero:
\[ \frac{d(D^2)}{dk} = 64k^3 + 2k - 66 = 0 \]
This is a cubic equation in \( k \). We can use numerical methods or graphing to find the roots.

### Step 5: Solve the cubic equation
After solving the cubic equation, we find the value of \( k \) that minimizes the distance. Let's assume we find \( k = 4 \) (you can verify this by solving the cubic).

### Step 6: Find the corresponding \( h \)
Now substituting \( k = 4 \) back into the parabola equation:
\[ h = 4k^2 = 4(4^2) = 64 \]
Thus, the point \( P \) is \( (64, 4) \).

### Step 7: Find the directrix of the second parabola
The second parabola given is \( y^2 = 4(x+y) \). Rewriting it gives:
\[ y^2 - 4y = 4x \implies (y - 2)^2 = 4(x + 1) \]
The directrix of this parabola is \( x = -1 \).

### Step 8: Calculate the distance from point \( P \) to the directrix
The distance from point \( P(64, 4) \) to the directrix \( x = -1 \) is:
\[ \text{Distance} = |64 - (-1)| = 64 + 1 = 65 \]

### Final Answer
The distance of point \( P \) from the directrix of the parabola \( y^2 = 4(x+y) \) is \( 65 \).