Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=\frac{3}{4}, \quad r \sin \theta=5 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{15}{4 + 3\sin\theta} \).

Step 1: Understand the Conic Section

The given eccentricity is \( e = \frac{3}{4} \) which indicates that the conic is an ellipse, as \( e < 1 \). The conic section will have its focus at the pole (origin) and its directrix is given as \( r \sin \theta = 5 \), which is a vertical line at distance 5 from the pole.

Step 2: Identify the Form of Polar Equation

For a conic section with focus at the pole, the polar equation can be given by \( r = \frac{ed}{1 + e\sin\theta} \) when the directrix is a vertical line. Here, \( e \) is the eccentricity and \( d \) is the distance from the origin to the directrix, which is 5.

Step 3: Substitute Values into Polar Equation

Since the directrix is \( r \sin \theta = 5 \), substitute \( e = \frac{3}{4} \) and \( d = 5 \) into the equation \( r = \frac{ed}{1 + e\sin\theta} \) to get:\[r = \frac{\frac{3}{4} \times 5}{1 + \frac{3}{4} \sin \theta}\]

Step 4: Simplify the Polar Equation

Simplifying the equation from Step 3:\[r = \frac{\frac{15}{4}}{1 + \frac{3}{4} \sin \theta} = \frac{15}{4 + 3\sin\theta}\]The polar equation for the conic is now simplified and complete.

Key Concepts

Conic Sections

Conic sections are crucial concepts in geometry that emerge when a plane intersects a cone. Depending on the angle and position of this intersection, different types of curves can be formed. These include circles, ellipses, parabolas, and hyperbolas. A useful way to identify the type of conic section formed is by its eccentricity.

• If eccentricity (\( e \)) equals 0, the conic is a circle.
• If \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), it is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

Conic sections have widespread applications, including planetary orbits and architectural designs. Understanding these shapes helps in exploring the world around us, from the way planets move to the structure of bridges and buildings.

Eccentricity

Eccentricity is a numerical value that defines the shape of a conic section. It determines how much the shape deviates from being circular. Eccentricity is particularly important as it dictates the nature of the conic:

• A value of \( e < 1 \) indicates an ellipse, which is the case in our original exercise where \( e = \frac{3}{4} \).
• A circle is a special case of an ellipse where \( e = 0 \).
• For values equal to one, the conic section is a parabola.
• For values greater than one, we see a hyperbola forming.

This mathematical characteristic is useful for understanding the geometry of the figure formed. In the context of polar coordinates, eccentricity allows us to derive the conic's equation relative to its directrix and focus.

Polar Equation

The polar equation of a conic section provides a way to describe the conic shape using polar coordinates, which involve a radius and an angle. This differs from the Cartesian system that uses x and y coordinates.For a conic section with focus at the pole, the polar equation is typically written as:\[ r = \frac{ed}{1 + e \sin \theta} \] where:

• \( e \) is the eccentricity.
• \( d \) is the distance from the pole to the directrix.

The exercise provided an eccentricity of \( \frac{3}{4} \) and a directrix represented by \( r \sin \theta = 5 \), leading to the simplified polar expression \( r = \frac{15}{4 + 3 \sin \theta} \).Having a grasp of how to set up and manipulate this equation is essential for applications in physics, engineering, and navigation systems where circular and elliptical paths are common.

Ellipse

An ellipse is a geometric shape formed when the eccentricity \( 0 < e < 1 \). It resembles an elongated circle, characterized by its two focal points. All the points on an ellipse have the property that the sum of the distances to the two foci is constant.In polar coordinates, the equation becomes increasingly significant as it helps in visualizing the ellipse with respect to a central focal point. In the given exercise, the value of \( e = \frac{3}{4} \) denotes an ellipse, which implies a stretched circle with a vertical orientation dictated by the directrix \( r \sin \theta = 5 \).Understanding ellipses is vital in various scientific fields, such as astronomy, where planets and satellites orbit in elliptical paths around stars. Moreover, this knowledge aids in crafting lenses and reflectors in optical designs.