Let `vecr` be position vector of variable point in cartesian plane OXY such that `vecr*(vecr+6hatj)=7` cuts the co-ordinate axes at four distinct points, then the area of the quadrilateral formed by joining these points is :

Options

A. `4sqrt(7)`
B. `6sqrt(7)`
C. `7sqrt(7)`
D. `8sqrt(7)`

Answer

To solve the problem step-by-step, we start by analyzing the given equation and determining the points where it intersects the coordinate axes.

### Step 1: Define the position vector
Let the position vector \(\vec{r}\) of the variable point in the Cartesian plane \(OXY\) be given by:
\[ \vec{r} = x \hat{i} + y \hat{j} \]

### Step 2: Substitute into the given equation
The equation provided in the problem is:
\[ \vec{r} \cdot (\vec{r} + 6 \hat{j}) = 7 \]
Substituting \(\vec{r}\) into the equation:
\[ (x \hat{i} + y \hat{j}) \cdot (x \hat{i} + (y + 6) \hat{j}) = 7 \]

### Step 3: Compute the dot product
Calculating the dot product:
\[ x^2 + y(y + 6) = 7 \]
This simplifies to:
\[ x^2 + y^2 + 6y = 7 \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ x^2 + y^2 + 6y - 7 = 0 \]
To express this in standard form, we complete the square for the \(y\) terms:
\[ x^2 + (y^2 + 6y + 9) - 9 - 7 = 0 \]
This simplifies to:
\[ x^2 + (y + 3)^2 = 16 \]

### Step 5: Identify the circle's properties
The equation \(x^2 + (y + 3)^2 = 4^2\) represents a circle with:
- Center: \((0, -3)\)
- Radius: \(4\)

### Step 6: Find the points of intersection with the axes
To find the points where the circle intersects the axes, we will set \(y = 0\) and \(x = 0\).

1. **Finding x-intercepts (set \(y = 0\))**:
\[ x^2 + (0 + 3)^2 = 16 \]
\[ x^2 + 9 = 16 \implies x^2 = 7 \implies x = \pm \sqrt{7} \]
Points: \((\sqrt{7}, 0)\) and \((- \sqrt{7}, 0)\)

2. **Finding y-intercepts (set \(x = 0\))**:
\[ 0 + (y + 3)^2 = 16 \]
\[ (y + 3)^2 = 16 \implies y + 3 = \pm 4 \]
This gives:
\[ y = 1 \quad \text{and} \quad y = -7 \]
Points: \((0, 1)\) and \((0, -7)\)

### Step 7: Identify the vertices of the quadrilateral
The four points where the circle intersects the axes are:
1. \((\sqrt{7}, 0)\)
2. \((- \sqrt{7}, 0)\)
3. \((0, 1)\)
4. \((0, -7)\)

### Step 8: Calculate the area of the quadrilateral
To find the area of the quadrilateral formed by these points, we can divide it into two triangles:
- Triangle A: \((-\sqrt{7}, 0)\), \((\sqrt{7}, 0)\), \((0, 1)\)
- Triangle B: \((-\sqrt{7}, 0)\), \((\sqrt{7}, 0)\), \((0, -7)\)

#### Area of Triangle A
Base = distance between \((- \sqrt{7}, 0)\) and \((\sqrt{7}, 0)\) = \(2\sqrt{7}\)
Height = \(1\) (from point \((0, 1)\))
\[ \text{Area}_A = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 2\sqrt{7} \times 1 = \sqrt{7} \]

#### Area of Triangle B
Base = distance between \((- \sqrt{7}, 0)\) and \((\sqrt{7}, 0)\) = \(2\sqrt{7}\)
Height = \(7\) (from point \((0, -7)\))
\[ \text{Area}_B = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 2\sqrt{7} \times 7 = 7\sqrt{7} \]

### Step 9: Total Area of the Quadrilateral
\[ \text{Total Area} = \text{Area}_A + \text{Area}_B = \sqrt{7} + 7\sqrt{7} = 8\sqrt{7} \]

### Final Answer
The area of the quadrilateral formed by joining these points is:
\[ \boxed{8\sqrt{7}} \]