Question

Use traces to sketch and identify the surface.

\(25{x^2} + 4{y^2} + {z^2} = 100\)

Short answer

The surface \({x^2} = {y^2} + 4{z^2}\) is an ellipsoid with the x-axis.

Step-by-step solution

standard equation of ellipsoid

Consider the given surface is\(25{x^2} + 4{y^2} + {z^2} = 100\).

The standard equation of ellipsoid\(x\)is\(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1\).

possible solution of the given surface equation

Rewrite the surface equation as\(25{x^2} + 4{y^2} + {z^2} = 100\).

Compare the standard form of the equation with\(25{x^2} + 4{y^2} + {z^2} = 100\)that satisfies the equation of an ellipsoid.

Divide the equation by 100 on both sides to make right hand side as 1.

\(\begin{array}{l}\frac{{25{x^2} + 4{y^2} + {z^2}}}{{100}} = \frac{{100}}{{100}}\\\frac{{25{x^2}}}{{100}} + \frac{{4{y^2}}}{{100}} + \frac{{{z^2}}}{{100}} = 1\\\frac{{{x^2}}}{4} + \frac{{{y^2}}}{{25}} + \frac{{{z^2}}}{{100}} = 1\end{array}\)

Compare the standard form of the equation with \({x^2} = {y^2} + 4{z^2}\) that satisfies the equation of ellipsoid. Thus, the surface equation \(25{x^2} + 4{y^2} + {z^2} = 100\) is an ellipsoid.

plot the ellipsoid

Plot the ellipsoid surface \(25{x^2} + 4{y^2} + {z^2} = 100\) as shown below in Figure 1.

From Figure 1, it is observed that the surface is ellipsoid with centre at the origin.