Question

Find the coordinates of the points of intersection of the lines represented by \( x^2 - y^2 - 2x + 1 = 0 \).

Hint:

When solving quadratic equations involving two variables, look for ways to simplify or factor the equation, such as completing the square.

Answer 1

Step 1: Rewrite the given equation.
The given equation is: \[ x^2 - y^2 - 2x + 1 = 0 \] Rearranging the terms: \[ x^2 - 2x - y^2 + 1 = 0 \]

Step 2: Complete the square for the \(x\)-terms.
To complete the square for the \(x\)-terms, add and subtract \(1\): \[ (x^2 - 2x + 1) - y^2 = 0 \] This simplifies to: \[ (x - 1)^2 - y^2 = 0 \]

Step 3: Recognize the difference of squares.
We now have a difference of squares: \[ (x - 1)^2 = y^2 \] Taking square roots on both sides: \[ x - 1 = \pm y \] So, we have two equations: \[ x - 1 = y \text{or} x - 1 = -y \]

Step 4: Solve the system of equations.
For \(x - 1 = y\), we have: \[ x = y + 1 \] For \(x - 1 = -y\), we have: \[ x = -y + 1 \]

Step 5: Conclude the solutions.
Thus, the points of intersection are given by the equations \( x = y + 1 \) and \( x = -y + 1 \).

Final Answer: The points of intersection are the solutions to the equations \( x = y + 1 \) and \( x = -y + 1 \).