General Polar Equations of Conics

Conic sections (circles, ellipses, parabolas, and hyperbolas) can be described using polar coordinates. A conic section is defined as the locus of points where the ratio of the distance to a fixed point (the focus) and the distance to a fixed line (the directrix) is a constant value called the eccentricity. In polar coordinates, this relationship leads to elegant and unified equations.

Understanding Eccentricity (e)

The eccentricity is a fundamental property that determines the shape of a conic section:

• If e = 0, the conic is a circle.
• If 0 < e < 1, the conic is an ellipse.
• If e = 1, the conic is a parabola.
• If e > 1, the conic is a hyperbola.

Deriving the Polar Equation

Let's place the focus of the conic at the origin (pole) of our polar coordinate system. Let the directrix be a line. We can choose the directrix to be a vertical or horizontal line for simplicity.

Consider a point P with polar coordinates (r, θ). The distance from the pole (focus) to P is simply r.

Now, let's define the directrix.

Case 1: Directrix is a vertical line x = d

If the directrix is the vertical line x = d, the distance from point P(r, θ) to this line is |d - r cos(θ)|.

By the definition of a conic, the ratio of the distance to the focus (r) to the distance to the directrix (|d - r cos(θ)|) is equal to the eccentricity e.

So, we have: r / |d - r cos(θ)| = e

This gives us: r = e |d - r cos(θ)|

Assuming d is positive (directrix to the right of the focus) and we are considering points where r cos(θ) < d, we can remove the absolute value: r = e (d - r cos(θ))

Now, we rearrange this equation to solve for r: r = ed - er cos(θ) r + er cos(θ) = ed r(1 + e cos(θ)) = ed r = ed / (1 + e cos(θ))

This is the general polar equation of a conic with a focus at the origin and a vertical directrix x = d.

Case 2: Directrix is a horizontal line y = d

If the directrix is the horizontal line y = d, the distance from point P(r, θ) to this line is |d - r sin(θ)|.

Following the same logic: r / |d - r sin(θ)| = e r = e |d - r sin(θ)|

Assuming d is positive (directrix above the focus) and r sin(θ) < d: r = e (d - r sin(θ)) r = ed - er sin(θ) r + er sin(θ) = ed r(1 + e sin(θ)) = ed r = ed / (1 + e sin(θ))

This is the general polar equation for a conic with a focus at the origin and a horizontal directrix y = d.

Summary of General Polar Equations

The general polar equation of a conic section with a focus at the origin is given by:

r = ed / (1 ± e cos(θ)) (for a vertical directrix)

r = ed / (1 ± e sin(θ)) (for a horizontal directrix)

Here:

• r is the distance from the pole (focus).
• θ is the angle from the polar axis.
• e is the eccentricity.
• d is the distance from the focus to the directrix.

The sign in the denominator depends on the position of the directrix relative to the focus.

Sign Conventions for the Denominator:

• 1 + e cos(θ): Directrix is x = d (vertical, to the right of the pole).
• 1 - e cos(θ): Directrix is x = -d (vertical, to the left of the pole).
• 1 + e sin(θ): Directrix is y = d (horizontal, above the pole).
• 1 - e sin(θ): Directrix is y = -d (horizontal, below the pole).

Analyzing Specific Conics using Polar Equations

1. Circle

A circle has e = 0. If we substitute e = 0 into the general polar equation, we get:

r = (0 * d) / (1 ± 0 * cos(θ)) = 0 / 1 = 0

This doesn't seem right. The standard polar equation for a circle is r = constant. The general polar equation form derived above assumes e > 0. A circle is a limiting case where the focus can be anywhere, and the directrix is infinitely far away.

However, if we consider a circle with a focus at the origin and a "directrix" at infinity, the definition of a conic breaks down. For practical purposes in this context, circles are usually represented by r = constant in polar coordinates, which is centered at the pole. If the circle is not centered at the pole, its polar equation is more complex. For the scope of the general polar equation of conics (where e is involved), we typically consider e > 0.

2. Ellipse

For an ellipse, 0 < e < 1.

Example: r = 4 / (1 + 0.5 cos(θ))

Here, e = 0.5 and ed = 4. So, d = 4 / 0.5 = 8.

This represents an ellipse with eccentricity 0.5, focus at the origin, and a vertical directrix x = 8.

Let's find some points:

• When θ = 0: r = 4 / (1 + 0.5 * 1) = 4 / 1.5 = 8/3. This is the perihelion (closest point to focus).
• When θ = π: r = 4 / (1 + 0.5 * (-1)) = 4 / 0.5 = 8. This is the aphelion (farthest point from focus).
• When θ = π/2 or 3π/2: r = 4 / (1 + 0) = 4. This is the semi-latus rectum.

The distance between perihelion and aphelion is 8 - 8/3 = 16/3. This is the major axis length (2a). So, a = 8/3.

The relationship for an ellipse is a = ed / (1 - e²). Let's check: a = (0.5 * 8) / (1 - 0.5²) = 4 / (1 - 0.25) = 4 / 0.75 = 4 / (3/4) = 16/3. This is the length of the major axis, not a.

Let's re-evaluate. The distance from focus to vertex along the major axis is r. For θ = 0, r = 8/3. This is one vertex. For θ = π, r = 8. This is the other vertex. The distance between vertices is 8 - 8/3 = 16/3. This is the length of the major axis, 2a. So, a = 8/3.

The distance from the center of the ellipse to the focus is c. For an ellipse, c = ae. Here, e = 0.5. So, c = (8/3) * 0.5 = 4/3. The center of the ellipse is at a distance a - c from the vertex closest to the focus. Center's distance from focus = a - c = 8/3 - 4/3 = 4/3. The center is at (4/3, 0) in Cartesian coordinates if the major axis is along the x-axis. The directrix is at x = 8. Distance from center to directrix is 8 - 4/3 = 20/3. For an ellipse, distance from center to directrix is a/e. a/e = (8/3) / 0.5 = (8/3) / (1/2) = 16/3. This does not match 20/3.

Let's reconsider the formula r = ed / (1 + e cos(θ)). The distance from the focus (pole) to the directrix is d. The equation represents a conic with focus at origin and directrix x = d. In our example, r = 4 / (1 + 0.5 cos(θ)). Here ed = 4 and e = 0.5. So d = 8. The directrix is x = 8. The vertex at θ = 0 is at r = 8/3. This vertex is at (8/3, 0). The vertex at θ = π is at r = 8. This vertex is at (-8, 0). The distance between these vertices is 8 - (-8/3) = 8 + 8/3 = 32/3. This is the major axis length, 2a. So, a = 16/3. The focus is at the origin (0,0). The vertices are at (-8, 0) and (8/3, 0). The center of the ellipse is the midpoint of the vertices: ((-8 + 8/3) / 2, 0) = ((-24/3 + 8/3) / 2, 0) = (-16/3 / 2, 0) = (-8/3, 0). The distance from the center (-8/3, 0) to the focus (0,0) is c = |-8/3 - 0| = 8/3. Check eccentricity: e = c / a = (8/3) / (16/3) = 8/16 = 1/2 = 0.5. This matches. Check distance from center to directrix: a/e = (16/3) / 0.5 = (16/3) / (1/2) = 32/3. The directrix is x = 8. The center is at x = -8/3. The distance is |8 - (-8/3)| = |8 + 8/3| = |24/3 + 8/3| = 32/3. This matches. So, the equation r = 4 / (1 + 0.5 cos(θ)) represents an ellipse with a = 16/3, e = 0.5, focus at origin, center at (-8/3, 0), and directrix x = 8.

3. Parabola

For a parabola, e = 1.

Example: r = 3 / (1 + cos(θ))

Here, e = 1 and ed = 3. So, d = 3.

This represents a parabola with eccentricity 1, focus at the origin, and a vertical directrix x = 3.

Let's find some points:

• When θ = 0: r = 3 / (1 + 1) = 3/2. This is the vertex.
• When θ = π: r = 3 / (1 - 1) = 3 / 0, which approaches infinity.
• When θ = π/2 or 3π/2: r = 3 / (1 + 0) = 3. This is the semi-latus rectum.

The vertex is at (3/2, 0) in Cartesian coordinates. The focus is at (0,0). The directrix is x = 3. The distance from the vertex (3/2, 0) to the focus (0,0) is 3/2. The distance from the vertex (3/2, 0) to the directrix x = 3 is |3 - 3/2| = 3/2. This confirms it's a parabola, as the vertex is equidistant from the focus and directrix.

The general form for a parabola with focus at the origin and directrix x = d is r = d / (1 + cos(θ)). The vertex is at (d/2, 0).

4. Hyperbola

For a hyperbola, e > 1.

Example 1: r = 2 / (1 - 2 cos(θ))

Here, e = 2 and ed = 2. So, d = 1.

This represents a hyperbola with eccentricity 2, focus at the origin, and a vertical directrix x = -1 (because of the minus sign).

Let's find some points:

• When θ = 0: r = 2 / (1 - 2 * 1) = 2 / (-1) = -2. This corresponds to a point in the opposite direction of θ = 0, so at (2, 0). This is a vertex.
• When θ = π: r = 2 / (1 - 2 * (-1)) = 2 / (1 + 2) = 2/3. This point is at (-2/3, 0). This is another vertex.
• When θ = π/2 or 3π/2: r = 2 / (1 - 0) = 2. These points are (0, 2) and (0, -2).

The vertices are at (2, 0) and (-2/3, 0). The distance between vertices is 2 - (-2/3) = 2 + 2/3 = 8/3. This is the transverse axis length, 2a. So, a = 4/3. The focus is at the origin (0,0). The directrix is x = -1. The distance from the focus to the directrix is d = 1. Check eccentricity: e = a/distance from center to vertex. We need the center. Center is the midpoint of vertices: ((2 + (-2/3)) / 2, 0) = ((6/3 - 2/3) / 2, 0) = (4/3 / 2, 0) = (2/3, 0). Distance from center (2/3, 0) to focus (0,0) is c = |2/3 - 0| = 2/3. Eccentricity e = c / a = (2/3) / (4/3) = 2/4 = 1/2. This does NOT match e = 2.

Let's re-examine the derivation and interpretation. The equation is r = ed / (1 ± e cos(θ)) or r = ed / (1 ± e sin(θ)). The derivation assumed r = e * (distance to directrix). If directrix is x = d (d>0), distance is |d - r cos(θ)|. If we assume d > r cos(θ), then r = e(d - r cos(θ)) => r(1 + e cos(θ)) = ed => r = ed / (1 + e cos(θ)). This form assumes the directrix is to the right of the focus (x = d). If the directrix is x = -d (d>0), distance is |-d - r cos(θ)| = |d + r cos(θ)|. If we assume d + r cos(θ) > 0, then r = e(d + r cos(θ)) => r = ed + er cos(θ) => r(1 - e cos(θ)) = ed => r = ed / (1 - e cos(θ)). This form has the directrix to the left of the focus.

Let's re-analyze Example 1: r = 2 / (1 - 2 cos(θ)). Here, e = 2. The form is r = ed / (1 - e cos(θ)). So, ed = 2. With e = 2, we get d = 1. This corresponds to a hyperbola with focus at the origin, eccentricity e = 2, and directrix x = -1.

Vertices: For θ = 0: r = 2 / (1 - 2) = -2. Point is (2, 0). For θ = π: r = 2 / (1 - 2(-1)) = 2 / 3. Point is (-2/3, 0). These are the vertices. The distance between them is 2 - (-2/3) = 8/3. So, 2a = 8/3, and a = 4/3. The focus is at (0,0). The vertices are at (2, 0) and (-2/3, 0). The center is the midpoint: ((2 - 2/3)/2, 0) = ((4/3)/2, 0) = (2/3, 0). Distance from center (2/3, 0) to focus (0,0) is c = |2/3 - 0| = 2/3. Eccentricity e = c / a = (2/3) / (4/3) = 2/4 = 1/2. Still not matching e=2.

There seems to be a confusion in interpreting d in the equation r = ed / (1 ± e cos(θ)). In the derivation, d was the distance from the focus to the directrix. Let's re-state the standard forms with focus at origin:

Standard Polar Equations of Conics (Focus at Origin)

In these standard forms, p is a constant related to the distance from the focus to the directrix. Specifically, if the directrix is x = D or y = D, then p = eD. Thus, the directrix is located at x = p/e or y = p/e (with appropriate signs). The term p is also known as the semi-latus rectum.

Let's re-analyze the previous examples using this standard form where p is the semi-latus rectum.

Re-analysis Example 1: Ellipse

r = 4 / (1 + 0.5 cos(θ))

Here, p = 4 and e = 0.5.

Since 0 < e < 1, it's an ellipse.

The form is 1 + e cos(θ), so the directrix is vertical and to the right of the focus.

The distance from the focus to the directrix is D = p/e = 4 / 0.5 = 8. So the directrix is x = 8.

The semi-latus rectum is p = 4. This is the distance from the focus to the curve along the line perpendicular to the major axis.

Vertices: θ = 0: r = 4 / (1 + 0.5) = 4 / 1.5 = 8/3. (Vertex V1) θ = π: r = 4 / (1 - 0.5) = 4 / 0.5 = 8. (Vertex V2)

The distance between vertices is 8 - 8/3 = 16/3. This is the major axis length, 2a. So, a = 8/3.

Focus is at (0,0). Vertices are at (8/3, 0) and (-8, 0). Center is midpoint: ((8/3 - 8)/2, 0) = ((8/3 - 24/3)/2, 0) = (-16/3 / 2, 0) = (-8/3, 0). Distance from center (-8/3, 0) to focus (0,0) is c = |-8/3 - 0| = 8/3. Check e = c / a = (8/3) / (8/3) = 1. This is incorrect. The eccentricity given was 0.5.

Let's re-check the definition of p. In the equation r = ed / (1 + e cos(θ)), d was the distance from focus to directrix. So ed was the numerator. If we rewrite r = p / (1 + e cos(θ)), then p = ed. So, the directrix is at distance d = p/e from the focus. In r = 4 / (1 + 0.5 cos(θ)), p = 4, e = 0.5. Distance to directrix d = p/e = 4 / 0.5 = 8. Directrix is x = 8. Semi-latus rectum is p = 4. Vertices are at r = p / (1+e) = 4 / (1+0.5) = 4/1.5 = 8/3 and r = p / (1-e) = 4 / (1-0.5) = 4/0.5 = 8. These distances are from the focus. Vertex 1 (closest to focus): distance 8/3. Located at (8/3, 0). Vertex 2 (farthest from focus): distance 8. Located at (-8, 0). Major axis length 2a = 8 - 8/3 = 16/3. So a = 8/3. Center is midpoint of vertices: ((8/3 - 8)/2, 0) = (-8/3, 0). Distance from center (-8/3, 0) to focus (0,0) is c = |-8/3| = 8/3. Eccentricity e = c / a = (8/3) / (8/3) = 1. STILL WRONG.

Let's use the definition: distance from center to vertex is a. Distance from center to focus is c. Eccentricity e = c/a. The polar equation gives distances from the focus. Vertex at θ=0: r = 8/3. This is distance from focus. Let this be one vertex position, say x_v1. Vertex at θ=π: r = 8. This is distance from focus. Let this be the other vertex position, say x_v2. The major axis lies along the line connecting these two points. The focus is at the origin. The directrix is at x = 8. The vertex closest to the directrix is at r = 8/3. This vertex is at (8/3, 0). The vertex farthest from the directrix is at r = 8. This vertex is at (-8, 0). The distance between vertices is 8 - (-8/3) = 8 + 8/3 = 32/3. This is 2a. So a = 16/3. The center is the midpoint of the vertices: ((8/3 - 8)/2, 0) = (-8/3, 0). The distance from the center (-8/3, 0) to the focus (0,0) is c = |-8/3 - 0| = 8/3. Eccentricity e = c / a = (8/3) / (16/3) = 8/16 = 1/2. THIS MATCHES! So, r = 4 / (1 + 0.5 cos(θ)) is indeed an ellipse with e = 0.5, a = 16/3, center at (-8/3, 0), focus at (0,0), and directrix x = 8. The semi-latus rectum is p = 4. Let's check if p = a(1 - e²). a(1 - e²) = (16/3) * (1 - 0.5²) = (16/3) * (1 - 0.25) = (16/3) * (0.75) = (16/3) * (3/4) = 16/4 = 4. Yes, it matches.

Re-analysis Example 2: Parabola

r = 3 / (1 + cos(θ))

Here, p = 3 and e = 1.

Since e = 1, it's a parabola.

The form is 1 + e cos(θ), so the directrix is vertical and to the right of the focus.

Distance from focus to directrix d = p/e = 3 / 1 = 3. Directrix is x = 3.

Semi-latus rectum p = 3.

Vertex: θ = 0: r = 3 / (1 + 1) = 3/2. This is the vertex position along the positive x-axis. Vertex at (3/2, 0). θ = π: r -> infinity.

The vertex is at (3/2, 0). The focus is at (0,0). The directrix is x = 3. Distance from vertex to focus = 3/2. Distance from vertex to directrix = |3 - 3/2| = 3/2. This confirms it's a parabola.

Re-analysis Example 3: Hyperbola

r = 2 / (1 - 2 cos(θ))

This equation does not fit the standard form r = p / (1 ± e cos(θ)) or r = p / (1 ± e sin(θ)) directly because the coefficient of cos(θ) is not 1.

To use the standard form, we must divide the numerator and denominator by the coefficient of cos(θ): r = (2/2) / ((1/2) - (2/2) cos(θ)) r = 1 / (0.5 - cos(θ))

This still doesn't fit the standard form where the denominator starts with 1. We need to factor out the coefficient of cos(θ) from the entire denominator. r = 1 / [ -1 * (cos(θ) - 0.5) ] r = -1 / (cos(θ) - 0.5) r = 1 / (0.5 - cos(θ))

Let's rewrite the original equation: r = 2 / (1 - 2 cos(θ)). We need the denominator to be in the form (1 ± e cos(θ)). Divide numerator and denominator by 2: r = (2/2) / ((1/2) - (2/2) cos(θ)) r = 1 / (0.5 - cos(θ)) This is still not in the standard form.

Let's go back to the initial derivation: r = e * (distance to directrix). If directrix is x = -d (d>0), distance is | -d - r cos(θ) | = | d + r cos(θ) |. Assuming d + r cos(θ) > 0, then r = e(d + r cos(θ)). r = ed + er cos(θ) r(1 - e cos(θ)) = ed r = ed / (1 - e cos(θ)) In this form, e is the eccentricity and d is the distance from the focus (pole) to the directrix.

Consider r = 2 / (1 - 2 cos(θ)). Comparing with r = ed / (1 - e cos(θ)): We MUST have e = 2. Then ed = 2. So, 2 * d = 2, which means d = 1. This represents a hyperbola with focus at the origin, eccentricity e = 2, and directrix x = -1.

Vertices: For θ = 0: r = 2 / (1 - 2) = -2. This point is at (2, 0). For θ = π: r = 2 / (1 - 2(-1)) = 2 / 3. This point is at (-2/3, 0).

The vertices are at (2, 0) and (-2/3, 0). The distance between vertices is 2 - (-2/3) = 8/3. This is the transverse axis length, 2a. So, a = 4/3.

The focus is at (0,0). The vertices are at (2, 0) and (-2/3, 0). The center is the midpoint of the vertices: ((2 - 2/3)/2, 0) = ((4/3)/2, 0) = (2/3, 0).

Distance from center (2/3, 0) to focus (0,0) is c = |2/3 - 0| = 2/3.

Check eccentricity: e = c / a = (2/3) / (4/3) = 2/4 = 1/2. This again does not match the eccentricity e = 2 derived from the equation form.

What is wrong? The interpretation of the equation form r = ed / (1 ± e cos(θ)) requires that the coefficient of cos(θ) in the denominator IS the eccentricity e.

Let's re-state the fundamental relationship: distance to focus = e * distance to directrix. Let focus be at origin. Let directrix be x = k. Point P is (r, θ). Distance to focus is r. Distance to directrix is |k - r cos(θ)|. So, r = e |k - r cos(θ)|.

Case A: Directrix is x = d, where d > 0. r = e |d - r cos(θ)|. If we assume d > r cos(θ) (i.e., point is to the left of the directrix relative to the focus), then r = e(d - r cos(θ)) r = ed - er cos(θ) r(1 + e cos(θ)) = ed r = ed / (1 + e cos(θ)). Here, e is the eccentricity, d is the distance from focus to directrix.

Case B: Directrix is x = -d, where d > 0. r = e |-d - r cos(θ)| = e |d + r cos(θ)|. If we assume d + r cos(θ) > 0 (i.e., point is to the right of the directrix relative to the focus), then r = e(d + r cos(θ)) r = ed + er cos(θ) r(1 - e cos(θ)) = ed r = ed / (1 - e cos(θ)). Here, e is the eccentricity, d is the distance from focus to directrix.

So, in the forms r = ed / (1 ± e cos(θ)) or r = ed / (1 ± e sin(θ)), the coefficient of cos(θ) or sin(θ) IS the eccentricity e, and the numerator ed is the product of eccentricity and the distance from focus to directrix.

Let's re-re-analyze r = 2 / (1 - 2 cos(θ)). This MUST be of the form ed / (1 - e cos(θ)). Therefore, e = 2. And ed = 2. Substituting e=2 into ed=2 gives 2*d = 2, so d = 1. This means: Hyperbola, focus at origin, eccentricity e = 2, directrix x = -1.

Let's calculate the properties based on these parameters: Focus F = (0,0). Directrix L: x = -1. Eccentricity e = 2. Vertices are points V such that distance(V, F) = e * distance(V, L). Let V = (x, 0) on the x-axis. Distance(V, F) = |x|. Distance(V, L) = |x - (-1)| = |x + 1|. So, |x| = 2 * |x + 1|. Case 1: x = 2(x + 1) => x = 2x + 2 => -x = 2 => x = -2. Vertex V1 = (-2, 0). Case 2: x = -2(x + 1) => x = -2x - 2 => 3x = -2 => x = -2/3. Vertex V2 = (-2/3, 0). The vertices are at (-2, 0) and (-2/3, 0). The transverse axis length is 2a = |-2 - (-2/3)| = |-2 + 2/3| = |-4/3| = 4/3. So, a = 2/3. The center is the midpoint of the vertices: ((-2 - 2/3)/2, 0) = ((-8/3)/2, 0) = (-4/3, 0). Distance from center (-4/3, 0) to focus (0,0) is c = |-4/3 - 0| = 4/3. Check eccentricity: e = c / a = (4/3) / (2/3) = 4/2 = 2. THIS MATCHES!

So, the equation r = 2 / (1 - 2 cos(θ)) represents a hyperbola with e = 2, a = 2/3, center at (-4/3, 0), focus at (0,0), and directrix x = -1.

The issue was in interpreting the vertices from the polar equation directly. The polar equation gives the distance from the focus. For θ = 0, r = -2. This means the point is 2 units away from the focus in the direction opposite to θ=0 (i.e., along the negative x-axis). So the point is (-2, 0). This is vertex V1. For θ = π, r = 2/3. This means the point is 2/3 units away from the focus in the direction of θ=π (i.e., along the negative x-axis). So the point is (-2/3, 0). This is vertex V2. This interpretation is consistent.

Converting Between Polar and Cartesian Forms

We use the relations: x = r cos(θ) y = r sin(θ) r² = x² + y²

Let's convert r = 4 / (1 + 0.5 cos(θ)) to Cartesian form. Multiply by (1 + 0.5 cos(θ)): r(1 + 0.5 cos(θ)) = 4 r + 0.5 r cos(θ) = 4 Substitute r = sqrt(x² + y²) and r cos(θ) = x: sqrt(x² + y²) + 0.5 x = 4 sqrt(x² + y²) = 4 - 0.5 x Square both sides: x² + y² = (4 - 0.5 x)² x² + y² = 16 - 2 * 4 * 0.5 x + (0.5 x)² x² + y² = 16 - 4x + 0.25 x² Rearrange terms to group x² and y²: x² - 0.25 x² + y² + 4x - 16 = 0 0.75 x² + y² + 4x - 16 = 0 Multiply by 4 to clear decimals: 3x² + 4y² + 16x - 64 = 0

This is the Cartesian equation of the ellipse. Let's check the center. Complete the square for x terms: 3(x² + (16/3)x) + 4y² = 64 3(x² + (16/3)x + (8/3)²) + 4y² = 64 + 3(8/3)² 3(x + 8/3)² + 4y² = 64 + 3(64/9) 3(x + 8/3)² + 4y² = 64 + 64/3 3(x + 8/3)² + 4y² = (192 + 64) / 3 = 256 / 3 Divide by 256/3: [3(x + 8/3)²] / (256/3) + [4y²] / (256/3) = 1 (x + 8/3)² / (256/9) + y² / (64/3) = 1 This is the standard form of an ellipse (x-h)²/a² + (y-k)²/b² = 1. Center (h, k) = (-8/3, 0). This matches our previous calculation. a² = 256/9 => a = 16/3. Matches. b² = 64/3 => b = 8/√3 = 8√3 / 3.

Applications of Polar Equations of Conics

Polar equations are particularly useful for describing orbits in astronomy. Kepler's first law states that planets orbit the Sun in elliptical paths, with the Sun at one focus. The polar equation of a conic section, with the focus at the origin, is the natural way to represent these orbits.

For example, the orbit of a comet or planet can be described by an equation of the form: r(θ) = p / (1 + e cos(θ)) where:

• r is the distance from the Sun (focus) to the celestial body.
• θ is the angle representing the position in the orbit.
• p is the semi-latus rectum, related to the closest approach distance (perihelion).
• e is the eccentricity of the orbit (e < 1 for ellipses).

By analyzing the values of p and e, astronomers can determine the shape and size of the orbit, the closest and farthest points from the Sun, and predict the celestial body's future path.