Question

For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$r=\frac{3}{413\sin \theta }$$

Short answer

The eccentricity is 413, and the conic is a hyperbola.

Step-by-step solution

Identify the polar equation format

The given polar equation is $r=\frac{3}{413\sin \theta }$. This equation is in the standard form of a polar conic $r=\frac{ed}{1+e\sin \theta }$, with $\theta$ as the angle, $e$ as the eccentricity, and $d$ as a constant.

Relate the equation to the standard form

Compare the given equation $r=\frac{3}{413\sin \theta }$ to the standard form $r=\frac{ed}{1+e\sin \theta }$. Notice that it has the form $r=\frac{3}{413\sin \theta }=\frac{ed}{e\sin \theta }$, implying $3=ed$ and $e\sin \theta$ matches $413\sin \theta$.

Solve for the eccentricity $e$

From the equation comparison, identify that the coefficient of $\sin \theta$ in the denominator gives $e$ directly. Hence, $e=413$ because the denominator directly affects the value of $e\sin \theta$.

Identify the conic based on the eccentricity

The type of conic section is determined by the value of eccentricity $e$:- A circle corresponds to $e=0$.- An ellipse has $0<e<1$.- A parabola has $e=1$.- A hyperbola has $e>1$.Here, $e=413$, which is greater than 1, indicating the conic is a hyperbola.

Key concepts

Eccentricity

Eccentricity, often denoted as $e$, is a crucial value used to classify the type of conic sections. It essentially measures how much a conic section deviates from being circular. Different ranges of eccentricity define different conic shapes.

• If $e=0$, the conic is a circle. This is because a circle is perfectly balanced around its center.
• When $0<e<1$, the conic forms an ellipse. Ellipses are stretched circles, so they have an eccentricity less than 1.
• For $e=1$, the conic is a parabola, which looks like a U-shaped curve.
• If $e>1$, the conic corresponds to a hyperbola, which consists of two separate curves.

Therefore, eccentricity quickly answers which shape a conic will take when observed through polar coordinates. In the exercise given, when $e=413$, it is unequivocally a hyperbola because its eccentricity is much greater than 1.

Conic Identification

Identifying conic sections involves recognizing the shape an equation represents. This exercise helps us place the polar equation within one of these classic structures.

The polar equation given is $r=\frac{3}{413\sin \theta }$. By transforming this equation into the standard polar form $r=\frac{ed}{1+e\sin \theta }$, we can deduce aspects like eccentricity and direct our identification of the conic section.

• For the given problem, comparing forms, specifically the denominators, tells us the eccentricity $e=413$.
• Finding $e>1$ reveals the conic is a hyperbola.

Through this simple substitution and coefficient comparison method, students learn how to quickly dissect and categorize polar equations into their respective conics.

Polar Coordinates

Polar coordinates offer a fascinating way to describe positions on a plane using a distance and an angle. Unlike traditional Cartesian coordinates $(x,y)$ which focus on horizontal and vertical positioning, polar coordinates pin down a point with a radius $r$ and angle $\theta$.

• The radius $r$ specifies how far from the origin a point is located.
• The angle $\theta$ indicates the direction relative to the positive $x$-axis. This is often measured counterclockwise.

This system is ideal for modeling circular or angular shapes, making it perfect for conics like circles, ellipses, and hyperbolas. In our exercise, the equation $r=\frac{3}{413\sin \theta }$ demonstrates the power of polar coordinates to express these shapes succinctly. Students see how polar coordinates easily handle the curves of a hyperbola by relating distance and angle in such equations.