The equation of pair of lines joining origin to the points of intersection of `x^2 + y^2 =9` and `x + y = 3`

Options

A. `x^(2)+(3-x)^(2)=9`
B. `xy=0`
C. `(3+y)^(2)+y^(2)=9`
D. `(x-y)^(2)=9`

Explanation

To find the equation of the pair of lines joining the origin to the points of intersection of the equations \(x^2 + y^2 = 9\) and \(x + y = 3\), we can follow these steps:

### Step 1: Find the Points of Intersection
We need to solve the two equations simultaneously to find the points of intersection.

1. The first equation is:
\[ x^2 + y^2 = 9 \]
2. The second equation is:
\[ x + y = 3 \quad \Rightarrow \quad y = 3 - x \]

Now, substitute \(y\) from the second equation into the first equation:
\[ x^2 + (3 - x)^2 = 9 \]

### Step 2: Expand and Simplify
Expanding the equation:
\[ x^2 + (3 - x)^2 = 9 \]
\[ x^2 + (9 - 6x + x^2) = 9 \]
\[ 2x^2 - 6x + 9 - 9 = 0 \]
\[ 2x^2 - 6x = 0 \]

### Step 3: Factor the Equation
Factoring out the common term:
\[ 2x(x - 3) = 0 \]

### Step 4: Solve for x
Setting each factor to zero gives:
1. \(2x = 0 \quad \Rightarrow \quad x = 0\)
2. \(x - 3 = 0 \quad \Rightarrow \quad x = 3\)

### Step 5: Find Corresponding y Values
Now, substitute these values back into \(y = 3 - x\) to find the corresponding \(y\) values:
1. For \(x = 0\):
\[ y = 3 - 0 = 3 \quad \Rightarrow \quad (0, 3) \]
2. For \(x = 3\):
\[ y = 3 - 3 = 0 \quad \Rightarrow \quad (3, 0) \]

### Step 6: Write the Equation of the Pair of Lines
The points of intersection are \((0, 3)\) and \((3, 0)\). The equation of the pair of lines joining the origin to these points can be expressed as:
\[ \frac{x}{0} + \frac{y}{3} = 1 \quad \text{and} \quad \frac{x}{3} + \frac{y}{0} = 1 \]

However, we can also express this in the form of the product of linear factors:
\[ y = mx \quad \text{for each point} \]

The equation of the pair of lines can be given by:
\[ xy = 9 \]

### Final Answer
Thus, the equation of the pair of lines joining the origin to the points of intersection is:
\[ xy = 9 \]