Question

Reduce the equation to one of the standard forms, classify the surface, and sketch it.

\(4{x^2} + {y^2} + 4{z^2} - 4y - 24z + 36 = 0\)

Short answer

The surface equation \(4{x^2} + {y^2} + 4{z^2} - 4y - 24z + 36 = 0\) represents an ellipsoid.

Step-by-step solution

Standard form of an ellipsoid

The standard equation of ellipsoid with centre at\((h,k,l)\)is,\(\frac{{{{(x - h)}^2}}}{{{a^2}}} - \frac{{{{(y - k)}^2}}}{{{b^2}}} + \frac{{{{(z - l)}^2}}}{{{c^2}}} = 1\).

Rewrite the given equation and compare with equation of ellipsoid

The given surface equation is\(4{x^2} + {y^2} + 4{z^2} - 4y - 24z + 36 = 0\).

Rearrange the above equation as follows.

Further simplify the equation

\(\begin{array}{l}\frac{{4{x^2}}}{4} + \frac{{{{(y - 2)}^2}}}{4} + \frac{{4{{(z - 3)}^2}}}{4} = \frac{4}{4}\\{x^2} + \frac{{{{(y - 2)}^2}}}{4} + {(z - 3)^2} = 1\end{array}\)

Compare the above equation with the standard equation of ellipsoid.

Observe that, the above equation is same as the standard equation of ellipsoid where\(a = 1,b = 2\)and\(c = 1.\)

Thus, the surface equation \(4{x^2} + {y^2} + 4{z^2} - 4y - 24z + 36 = 0\) represents an ellipsoid.

Graph the ellipsoid

Use online graphing calculator and sketch the graph of \(4{x^2} + {y^2} + 4{z^2} - 4y - 24z + 36 = 0\)