If the coordinates of a point are (x, y) where `x^(2)+2x-3=0` and `y^(2)-6y-7=0`, then the sum of the squares of the distances of all such points from the origin is K where K =

Answer

To solve the problem step by step, we will first find the values of \(x\) and \(y\) from the given equations, then calculate the distances from the origin, and finally sum the squares of these distances.

### Step 1: Solve for \(x\)

We start with the equation:

\[ x^2 + 2x - 3 = 0 \]

To factor this quadratic equation, we look for two numbers that multiply to \(-3\) (the constant term) and add to \(2\) (the coefficient of \(x\)). The numbers \(3\) and \(-1\) satisfy these conditions.

Thus, we can factor the equation as:

\[ (x + 3)(x - 1) = 0 \]

Setting each factor to zero gives us:

\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

So the possible values for \(x\) are \(-3\) and \(1\).

### Step 2: Solve for \(y\)

Next, we solve the equation:

\[ y^2 - 6y - 7 = 0 \]

Again, we need to factor this quadratic. We look for two numbers that multiply to \(-7\) and add to \(-6\). The numbers \(-7\) and \(1\) work.

Thus, we can factor the equation as:

\[ (y - 7)(y + 1) = 0 \]

Setting each factor to zero gives us:

\[ y - 7 = 0 \quad \Rightarrow \quad y = 7 \]
\[ y + 1 = 0 \quad \Rightarrow \quad y = -1 \]

So the possible values for \(y\) are \(7\) and \(-1\).

### Step 3: Determine the Points

From the values of \(x\) and \(y\), we can form the following points:

1. \((1, 7)\)
2. \((1, -1)\)
3. \((-3, 7)\)
4. \((-3, -1)\)

### Step 4: Calculate the Distances from the Origin

The distance \(d\) from the origin \((0, 0)\) to a point \((x, y)\) is given by the formula:

\[ d = \sqrt{x^2 + y^2} \]

We need to find the square of the distances for each point:

1. For \((1, 7)\):
\[ d^2 = 1^2 + 7^2 = 1 + 49 = 50 \]

2. For \((1, -1)\):
\[ d^2 = 1^2 + (-1)^2 = 1 + 1 = 2 \]

3. For \((-3, 7)\):
\[ d^2 = (-3)^2 + 7^2 = 9 + 49 = 58 \]

4. For \((-3, -1)\):
\[ d^2 = (-3)^2 + (-1)^2 = 9 + 1 = 10 \]

### Step 5: Sum the Squares of the Distances

Now, we sum the squares of the distances:

\[ K = 50 + 2 + 58 + 10 = 120 \]

Thus, the value of \(K\) is:

\[ \boxed{120} \]