Orienting the Ellipse

Vertical Orientation

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An ellipse is in the vertical position if the major axis is vertical. The standard form of a vertical ellipse with center \((h, k)\) is:

\(\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1\)

Take note that \(a > b\); Distance \(a\) should be the denominator of the \((y-k)\) term.

We can expand this standard form to obtain its general form:

\(A x^2+B y^2+C x+D y+E=0\)

Like the horizontal ellipse, it is a polynomial of second degree. In this case, \(B>A\) for a vertical ellipse.

Summary

An ellipse can have three orientations: vertical, horizontal, or inclined. We base these on the orientation of the ellipse's major axis.

An ellipse is said to be horizontal if its major axis is oriented horizontally. If we have a horizontal ellipse with center \((h, k)\), the standard equation is: \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)

An ellipse is in the vertical position if its major axis is oriented vertically. The standard form of a vertical ellipse with center \((h, k)\) is: \(\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1\)

The general form of both orientations is \(A x^2+B y^2+C x+D y+E=0\). It is a horizontal ellipse if \(A>B\). If \(B<A\), then it is vertical.