Problem 62 - Thomas Calculus

Question

Find a polar equation for the parabola with focus \((0,0)\) and direc\(\operatorname{trix} r \cos (\theta-\pi / 2)=2\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Standard Formula

A parabola in polar coordinates with focus at the pole is described by the formula: \( r = \frac{ed}{1 + e\cos(\theta)} \), where \( e \) is the eccentricity and \( d \) is the perpendicular distance from the focus to the directrix.

Step 2: Understand Given Parameters

We are given that the directrix is represented as \(r \cos(\theta - \frac{\pi}{2}) = 2\). This simplifies to \( r\sin(\theta) = 2 \). The focus is at \((0,0)\), and we know that for a parabola, eccentricity \( e = 1 \). Hence, \( d = 2 \).

Step 3: Plug Into Formula

Substitute \( e = 1 \) and \( d = 2 \) into the standard formula. Since the directrix is \( r\sin(\theta) = 2 \), the polar equation becomes: \[ r = \frac{2}{1 + \sin(\theta)} \].

Step 4: Final Equation

Verify that substituting \( e = 1 \) and \( d = 2 \) into the formula results in the equation \( r = \frac{2}{1 + \sin(\theta)} \), which correctly represents the parabola with the given focus and directrix.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique system for representing points in a plane. Unlike the Cartesian coordinate system, which uses

• x and y values to locate points,
• polar coordinates use a radius and an angle.

The point is described as

• (r, \( \theta \)), where r is the distance from the origin,
• and \( \theta \) is the angle made with the positive x-axis.

This coordinate system is especially useful for solving problems involving circular or rotational symmetry.
When working with polar coordinates, it's important to remember how they differentiate from Cartesian coordinates, and how equations in polar form can help simplify certain geometric problems.

Parabola

A parabola is a curve where any point is equidistant from a:

• fixed point, known as the focus,
• and a fixed line, called the directrix.

In a polar coordinate system, a parabola's equation differs slightly from its Cartesian counterpart.
Polar equations for parabolas often involve measuring distances using the radius, r, and angles, \( \theta \). In this exercise, the focus is at the point (0,0) or the pole. This simplifies the parabolic equation, allowing it to be expressed neatly in polar form.

Eccentricity

Eccentricity, denoted by e, measures the deviation of a curve from a perfect circle. It is a crucial concept in defining conic sections like ellipses, parabolas, and hyperbolas.

• For a parabola, the eccentricity is always 1.
• This value indicates that the shape of the parabola will be open, unlike an ellipse which closes in on itself.

In the given exercise, understanding that the eccentricity is 1 directly helps in formulating the equation of the parabola.
It tells us that the parabola is equidistant in relation to its directrix and focus, which is why the equation can have such a simplified form.

Directrix

The directrix is a crucial component in the definition of a parabola. It is a fixed line that works together with the focus

• to help shape the curve,
• ensuring that the distance from any point on the parabola to the focus equals the distance to the directrix.

In the realm of polar coordinates, the directrix can influence the equation form of conic sections like the parabola.
For this exercise, the directrix was given as r sin(\( \theta \)) = 2. This expression helps to set the perpendicular distance d from the focus.Directrix and the Equation:

• The simplified equation provides \( d = 2 \),
• which directly informs our calculations in the final equation of the parabola.

Understanding the role of the directrix ensures the correct setup of the equation for conics, pivotal for their geometric representation.