Let `vec(r)` is a positive vector of a variable pont in cartesian OXY plane such that `vecr.(10hatj-8hati-vecr)=40` and `p_1=max{|vecr+2hati-3hatj|^2},p_2=min{|vecr+2hati-3hatj|^2}`. Then `p_1 + p_2` ​is equal to

Options

A. 2
B. 10
C. 18
D. 5

Explanation

To solve the given problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Define the vector \(\vec{r}\)
Let \(\vec{r} = x \hat{i} + y \hat{j}\), where \(x\) and \(y\) are the coordinates of the point in the Cartesian OXY plane.

### Step 2: Use the given equation
We are given the equation:
\[ \vec{r} \cdot (10 \hat{j} - 8 \hat{i} - \vec{r}) = 40 \]
Substituting \(\vec{r}\) into the equation:
\[ (x \hat{i} + y \hat{j}) \cdot (10 \hat{j} - 8 \hat{i} - (x \hat{i} + y \hat{j})) = 40 \]
This simplifies to:
\[ (x \hat{i} + y \hat{j}) \cdot (10 \hat{j} - 8 \hat{i} - x \hat{i} - y \hat{j}) = 40 \]
Calculating the dot product:
\[ = x(-8 - x) + y(10 - y) = 40 \]
This leads to:
\[ -8x - x^2 + 10y - y^2 = 40 \]
Rearranging gives:
\[ x^2 + y^2 + 8x - 10y + 40 = 0 \]

### Step 3: Rewrite the equation
To find the center and radius of the circle, we complete the square:
\[ (x^2 + 8x) + (y^2 - 10y) + 40 = 0 \]
Completing the square:
\[ (x + 4)^2 - 16 + (y - 5)^2 - 25 + 40 = 0 \]
This simplifies to:
\[ (x + 4)^2 + (y - 5)^2 = 1 \]
Thus, the center of the circle is at \((-4, 5)\) and the radius is \(1\).

### Step 4: Determine \(p_1\) and \(p_2\)
We need to find \(p_1 = \max\{|\vec{r} + 2 \hat{i} - 3 \hat{j}|^2\}\) and \(p_2 = \min\{|\vec{r} + 2 \hat{i} - 3 \hat{j}|^2\}\).

#### Step 4.1: Find the distance from the point \((-2, 3)\)
The point \((-2, 3)\) is outside the circle. The distance from \((-2, 3)\) to the center \((-4, 5)\) is:
\[ d = \sqrt{((-2) - (-4))^2 + (3 - 5)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2} \]

#### Step 4.2: Calculate \(p_1\)
The maximum distance from the point to the circle is:
\[ d + \text{radius} = 2\sqrt{2} + 1 \]
Thus,
\[ p_1 = (2\sqrt{2} + 1)^2 = 4 \cdot 2 + 4\sqrt{2} + 1 = 8 + 4\sqrt{2} + 1 = 9 + 4\sqrt{2} \]

#### Step 4.3: Calculate \(p_2\)
The minimum distance from the point to the circle is:
\[ d - \text{radius} = 2\sqrt{2} - 1 \]
Thus,
\[ p_2 = (2\sqrt{2} - 1)^2 = 4 \cdot 2 - 4\sqrt{2} + 1 = 8 - 4\sqrt{2} + 1 = 9 - 4\sqrt{2} \]

### Step 5: Calculate \(p_1 + p_2\)
Now we can find \(p_1 + p_2\):
\[ p_1 + p_2 = (9 + 4\sqrt{2}) + (9 - 4\sqrt{2}) = 18 \]

### Final Answer
Thus, the value of \(p_1 + p_2\) is:
\[ \boxed{18} \]