Question

Find where the graph of the equation \[{y^2} + {z^2} = 0\] in the three-dimensional space lies.
A. \[x\]-axis
B. \[z\]-axis
C. \[y\]-axis
D. \[yz\]-plane

Answer 1

Hint: Here, an equation of the curve is given. First, check the values for which the given equation is true. Then, on the basis of the values of the equation, find the required answer. The square of any positive or negative number is always positive.

Complete step by step solution: The given equation of the graph is \[{y^2} + {z^2} = 0\].
We know that the square of any positive or negative number is positive.
So, \[{y^2} = - {z^2}\] is not possible for any number other than 0.
Thus, we get
\[{y^2} = 0\] and \[{z^2} = 0\]
\[ \Rightarrow y = 0\] and \[z = 0\]
\[ \Rightarrow \left( {x,y,z} \right) = \left( {x,0,0} \right)\]
Therefore, the graph of the equation \[{y^2} + {z^2} = 0\] lies on the \[x\]-axis.

Thus, Option (A) is correct.

Note: Remember the following information about the point \[\left( {x,y,z} \right)\] :
If \[y = 0\] and \[z = 0\], then the point \[\left( {x,y,z} \right)\] lies on the \[x\]-axis.
If \[x = 0\] and \[z = 0\], then the point \[\left( {x,y,z} \right)\] lies on the \[y\]-axis.
If \[x = 0\] and \[y = 0\], then the point \[\left( {x,y,z} \right)\] lies on the \[z\]-axis.