Question

Describe and sketch the surface of equation \(4{x^2} + {y^2} = 4\).

Short answer

The surface of equation \(4{x^2} + {y^2} = 4\) is described and sketched.

Step-by-step solution

Concept Used

Write the expression for an elliptic cylinder.

\(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\,\,\,\,\,\,\,\,(1)\)

Here,

r and\(y\)are coordinates, and

a and\(b\)are constants.

Represent the Surface of Equations

The equation \(4{x^2} + {y^2} = 4\) represents \(\left\{ {(x,y,z)\mid 4{x^2} + {y^2} = 4} \right\}\) and does not involve \(z\)-coordinates. Hence, set the z-plane equation as \(z = k\), where \(\hat k\) is a constant.

Re-arrange the equation \(4{x^2} + {y^2} = 4\).

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

Compare equations (1) and (2).

Therefore, the resultant curve of equation \({x^2} + {y^2} = 4\)forms an elliptic cylinder with infinitely many shifted duplicates of the same circular cylinder.

The rulings in the representation of \(4{x^2} + {y^2} = 4\) are parallel to the \(z\)-axis.

The sketch of equation \(4{x^2} + {y^2} = 4\) is shown below.