Question

The graph of equation \(y^2 + z^2 = 0\) in three dimensional space is

• A. YZ-plane
• B. Z-axis
• C. Y-axis
• D. X-axis

Hint:

Sum of squares = 0 $\Rightarrow$ each term = 0.

Answer 1

The Correct Option is D

The given equation is

\(y^2 + z^2 = 0\)

To analyze this equation, consider the following properties:

1. The equation \(y^2 + z^2 = 0\) implies that both \(y\) and \(z\) must simultaneously be zero. This is because the sum of squares of non-zero real numbers cannot equal zero.
2. Therefore, the only solution for this equation is \((y, z) = (0, 0)\).
3. This set of solutions can be visualized in 3D space as the set of all points along the \(x\)-axis where \(y = 0\) and \(z = 0\).
4. Therefore, the equation \(y^2 + z^2 = 0\) represents the X-axis in three-dimensional space.

Let's evaluate the other options:

• YZ-plane: This would require both \(x = 0\) and is not restricted by \(y\) or \(z\) equaling zero, which doesn't satisfy the equation.
• Z-axis: For the Z-axis, \(y = 0\), but \(z\) can be any value, which does not satisfy \(y^2 + z^2 = 0\) unless \(z = 0\) as well.
• Y-axis: Similarly, for the Y-axis, \(z = 0\), but \(y\) can be any value, which again does not satisfy the equation unless \(y = 0\).

Therefore, the correct answer is indeed the X-axis, as the equation defines the line in 3D space where both \(y\) and \(z\) are zero.