Polar Equations of Conics

Reading a Polar Conic Equation

A polar equation can describe a conic without displaying its center, axes, or directrix in the way a Cartesian equation might. In this tutorial, the main task is to recognize a specific family of equations and read their parameters correctly. The focus is at the pole, and the directrix is a horizontal or vertical line at distance \(d\) from it.

The key feature is that the same positive parameter \(e\) appears in two places: it multiplies \(d\) in the numerator, and it is the coefficient of the trigonometric term in the denominator. Matching those two appearances lets us recover both the eccentricity and the directrix distance.

The Two Standard Forms

When the directrix is the vertical line \(x=d\), the standard form is

When the directrix is the horizontal line \(y=d\), the standard form is

In both equations, \(e\) is the eccentricity and \(d\) is a positive distance. The trigonometric function indicates the orientation: cosine is paired with a vertical directrix, while sine is paired with a horizontal directrix. This follows the Cartesian relationships \(x=r\cos\theta\) and \(y=r\sin\theta\).

The plus sign places the directrix on the positive coordinate side: \(x=d\) in the cosine form and \(y=d\) in the sine form. If the denominator has a minus sign instead, the corresponding directrix is on the negative side, at \(x=-d\) or \(y=-d\). For example, \(r=\frac{ed}{1-e\cos\theta}\) corresponds to a vertical directrix on the negative \(x\)-axis.

Why the Trigonometric Term Identifies the Directrix

The coordinate conversions give \(x=r\cos\theta\) and \(y=r\sin\theta\). Consider the cosine form with positive \(r\). Multiplying by its denominator gives \(r+er\cos\theta=ed\), or \(r+ex=ed\). Thus \(r=e(d-x)\). On the side \(x\le d\), the quantity \(d-x\) is the distance to the directrix \(x=d\). The equation therefore relates distance from the focus to distance from that vertical directrix.

For the sine form, multiplication gives \(r+er\sin\theta=ed\), so \(r+ey=ed\), or \(r=e(d-y)\). On the side \(y\le d\), \(d-y\) is the distance to the horizontal directrix \(y=d\). This explains why cosine goes with a vertical line and sine with a horizontal line. The complete focus-directrix derivation, including the treatment of coordinate choices and signs, is a separate topic.

Worked Examples

A Reliable Recognition Process

When an equation is written with unfamiliar coefficients, first put it into a form that can be compared directly with a standard equation. For example, if the equation is \(r=\frac{A}{1+B\sin\theta}\), then \(B\) is the candidate for \(e\), and \(A\) is the candidate for \(ed\). If \(B>0\), then \(d=A/B\). Check that \(A=ed\) when you substitute the resulting values back.

These steps assume the equation is in the stated standard form and that the coefficients correspond to positive \(e\) and \(d\). If the coefficient has a minus sign, use its magnitude for \(e\) and account for the negative-side directrix. A zero numerator or zero trigonometric coefficient does not fit the nondegenerate standard forms with \(e>0\) and \(d>0\).

Common Mistake: Confusing \(e\) with \(ed\)

A frequent error is to treat the numerator as the eccentricity. In \(r=\frac{12}{1+3\cos\theta}\), one might incorrectly say \(e=12\), then take \(d=1\) because \(12=12\cdot1\). Although that choice reproduces the numerator, it fails to reproduce the denominator: \(1+e\cos\theta\) would become \(1+12\cos\theta\), not \(1+3\cos\theta\).

The denominator identifies \(e=3\). The numerator then gives \(ed=12\), so \(d=4\). Checking both places prevents the error: \(ed=3\cdot4=12\), and the denominator coefficient is \(3\). Both parts match.

What This Recognition Tells You—and What It Does Not

Once the equation is matched to a standard form, you know the focus is at the pole, you can identify the eccentricity, and you can locate the directrix. You also know the orientation of the directrix from the trigonometric function. This is enough to interpret the equation’s basic geometric parameters without expanding it into Cartesian coordinates.

Recognition alone does not require you to classify the conic by the value of \(e\), determine its complete graph, or find its asymptotes. Those questions involve additional analysis. Also note that at any angle where the denominator is zero, the displayed expression does not give a finite value of \(r\). Such angles must not be treated as ordinary points on the curve.

The central check is simple: the denominator coefficient gives \(e\), and the numerator must equal \(ed\). If both match, and the trigonometric function and sign have been interpreted correctly, the standard form has been recognized.