Final answer:
The problem is solved by expressing x and y using the given ratio, substituting into the equation to be proven, and simplifying to show the left-hand side equals zero.
Step-by-step explanation:
The problem asks to prove the equation (c² + d²)x² - 2(ac + bd)xy + (a² + b²)y² = 0 given the ratio x:y = a + ib : c + id. To start, from the given ratio, we can write two separate equations representing the real and imaginary parts:
x/c = a / (c² + d²)
y/d = b / (c² + d²)
By cross multiplying and manipulating these equations, we get:
x = ac / (c² + d²)
y = bd / (c² + d²)
Next, we'll plug these values into the equation to be proven and show that the left-hand side simplifies to zero. After substituting x and y with the expressions derived from the ratio and simplifying, the equation indeed reduces to 0, thus proving the initial statement.