Problem 30 - Thomas Calculus

Question

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$ e=1, \quad y=2 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{2}{1 + \sin \theta} \).

Step 1: Identify the type of conic section

The eccentricity given is \( e = 1 \). Since \( e = 1 \), the conic section is a parabola.

Step 2: Recognize the equation form

For a parabola with a focus at the origin and directrix \( y = k \), the polar equation is \( r = \frac{ed}{1 + e \sin \theta} \) if the directrix is horizontal. Here, \( d \) is the distance from the origin to the directrix, and \( e \) is the eccentricity.

Step 3: Identify the parameters for the equation

Since the directrix is \( y = 2 \), \( d = 2 \), and the eccentricity \( e = 1 \). Substitute these values into the polar equation formula for a parabola.

Step 4: Substitute values into the formula

Plug \( e = 1 \) and \( d = 2 \) into the formula \( r = \frac{ed}{1 + e \sin \theta} \) to get \( r = \frac{2 \cdot 1}{1 + 1 \sin \theta} = \frac{2}{1 + \sin \theta} \).

Step 5: Write the final polar equation

The polar equation of the parabola with given eccentricity and directrix is \( r = \frac{2}{1 + \sin \theta} \).

Key Concepts

Parabola

A parabola is a unique type of conic section distinguished by its shape, which resembles an open curve. This curve is symmetrical and looks like a U or an inverted U. Parabolas have some fascinating properties that make them important in both mathematics and physics.

• They are defined as the set of all points equidistant from a fixed point called the focus and a line called the directrix.
• In practical terms, a parabola's path can be seen in objects that follow a projectile trajectory, like a thrown ball.
• Parabolas are often used in designs, such as satellite dishes and headlights, to help reflect and focus light efficiently.

Understanding the nature of parabolas and their symmetry is crucial, especially when working with different coordinate systems like polar equations. The tasteful simplicity of a parabola beautifully showcases the harmony between algebra and geometry.

Eccentricity

Eccentricity is a fundamental concept when discussing conic sections, including parabolas, ellipses, and hyperbolas. It defines how "stretched" or "flattened" a conic section is.

• A conic section's eccentricity is denoted by the letter \( e \).
• For a circle, \( e = 0 \), meaning it is perfectly circular.
• An ellipse has an eccentricity where \( 0 < e < 1 \), giving it its oval shape.
• For a parabola, \( e = 1 \), which indicates its distinct property among conics of having a single focus and a directrix with all points being equidistant.
• A hyperbola has \( e > 1 \), which means it forms two open branches.

The eccentricity plays a pivotal role in defining the standard formula of conic sections in polar coordinates. Its value helps determine the geometric properties of the graph you will be plotting.

Directrix

The directrix is an integral element when studying conic sections such as parabolas. It is a fixed straight line that assists in defining a conic section.

• It acts as a reference line ensuring the distance from any point on the conic to the focus is proportional to its perpendicular distance from the directrix.
• For parabolas, the distance from the parabola to the focus is equal to the perpendicular distance from the parabola to the directrix.
• The position of the directrix affects the orientation and equation of the conic section. For instance, with a horizontal directrix at \( y = 2 \), the parabola opens downward or upward.

Understanding the directrix helps in plotting and identifying conics correctly, especially in exercises involving transformations between coordinate systems, such as rectangular to polar coordinates.

Polar Coordinates

Polar coordinates provide a different perspective on the Cartesian coordinate system. This system specifies a point by a distance from a reference point and an angle from a reference direction.

• In polar coordinates, a point is represented as \((r, \theta)\), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.
• They are particularly advantageous in contexts involving rotational symmetry or when dealing with circular or spiral patterns.
• Converting equations between polar and Cartesian systems can simplify problems, such as when identifying conic sections.
• A conic section like a parabola expressed in polar coordinates may appear as \( r = \frac{ed}{1 + e\sin\theta} \) for a horizontal directrix.

Using polar coordinates efficiently allows for broader applications in fields such as physics and engineering, where radial symmetry or rotation is significant.