The equation \( z^{2}+y^{2}-(\tan (x))^{2}=0 \) represents a surface of revolution. Around which axis was the profile curve revolved to generate this equation? \( x \)-axis \( y \)-axis Write the equation of the profile curve in the \( x z \)-plane.

The given equation is: \[ z^{2} + y^{2} - (\tan(x))^{2} = 0 \] This can be rewritten as: \[ z^{2} + y^{2} = \tan^{2}(x) \] ### Axis of Revolution To determine around which axis the profile curve was revolved, let's analyze the equation: - **Revolving Around the \( x \)-axis:** If we have a profile curve in the \( xz \)-plane described by \( z = f(x) \), revolving this curve around the \( x \)-axis generates a surface where every point on the curve traces a circle in the \( yz \)-plane. The equation of such a surface is: \[ y^{2} + z^{2} = f(x)^{2} \] Comparing this with the given equation \( y^{2} + z^{2} = \tan^{2}(x) \), we see a direct match. This indicates that the profile curve \( z = \tan(x) \) in the \( xz \)-plane was revolved around the \( x \)-axis to generate the given surface. - **Revolving Around the \( y \)-axis:** Revolving a curve around the \( y \)-axis would involve the \( x \) and \( z \) coordinates being expressed in terms of the radial distance from the \( y \)-axis, which doesn't align with the given equation structure. ### Equation of the Profile Curve in the \( xz \)-Plane Given that the surface is generated by revolving around the \( x \)-axis, the profile curve lies in the \( xz \)-plane and is described by: \[ z = \tan(x) \] ### Conclusion - **Axis of Revolution:** \( x \)-axis - **Profile Curve Equation in the \( xz \)-plane:** \( z = \tan(x) \) **Final Answer:** • The surface is obtained by rotating the profile curve about the x-axis. • In the xz-plane, the profile curve is z = tan x.