Problem 6 - Analytical Geometry: 2D and 3D

Question

If the circle \(r+2 a \cos \theta=0\) cuts the conic \(\frac{1}{r}=1+e \cos (\theta-\alpha)\) in four real points find the equation in \(r\) which determines the distances of these points from the pole. Also, show that if their algebraic sum equals \(2 a\) and the eccentricity of the conic is \(2 \cos \alpha\).

Step-by-Step Solution

Answer

Based on the given information and the step-by-step solution provided, create a short answer that determines the condition for the eccentricity of the conic: The condition for the eccentricity of the conic is \(e = 2\cos{\alpha}\), if and only if the sum of the distances from the points of intersection between the circle and the conic to the pole is equal to \(2a\).

Step 1: Identify the points of intersection between the circle and conic

To find the points of intersection, we'll equate the polar equations of the circle and the conic: \(r + 2a\cos{\theta} = \frac{1}{1 + e\cos{(\theta - \alpha)}}\) First, we will find the value of \(\cos{(\theta - \alpha)}\) in terms of r, a, and θ, by rearranging and solving for the cos term.

Step 2: Solve for the cos term

Multiply both sides by \((1 + e\cos{(\theta - \alpha)})\) to get rid of the fraction: \(r(1 + e\cos{(\theta - \alpha)}) = 1\) Now, subtract the \(2a\cos{\theta}\) term and solve for \(\cos{(\theta - \alpha)}\): \(e\cos{(\theta - \alpha)}= \frac{1-r}{r} - 2a\cos{\theta}\)

Step 3: Express the polar equation of the conic in terms of r

We have now found \(\cos{(\theta - \alpha)}\) and can substitute it back into the polar equation of the conic to express it in terms of r: \(\frac{1}{r} = 1+ \frac{1-r}{r} - 2a\cos{\theta}\) Solving for \(r\), we get: \(r = \frac{1}{3-2r - 2a\cos{\theta}}\) Though this equation includes the variable \(\theta\), it gives us the distances from the intersection points of the circle and the conic to the pole, as required.

Step 4: Find the condition for the eccentricity of the conic

Now, let's use the given condition that the algebraic sum of the distances equals \(2a\): \(r_1 + r_2 + r_3 + r_4 = 2a\) Replacing \(r_1\), \(r_2\), \(r_3\), and \(r_4\) with the expressions we found in Step 3 and the fact that \(\sum \cos{\theta} = 0\) since the points are symmetric, we get: \(2a = \frac{1}{3-2r_1 - 2a\cos{\theta_1}} + \frac{1}{3-2r_2 - 2a\cos{\theta_2}}+ \frac{1}{3-2r_3 - 2a\cos{\theta_3}} + \frac{1}{3-2r_4 - 2a\cos{\theta_4}}\) The numerator and denominator on the left must both be equal to one in order for the given condition to be satisfied: \(1 = 3-2r_1 - 2a\cos{\theta_1} + 3-2r_2 - 2a\cos{\theta_2}+ 3-2r_3 - 2a\cos{\theta_3} + 3-2r_4 - 2a\cos{\theta_4}\) Simplifying, we get: \(4 = (r_1 + r_2 + r_3 + r_4) = 2a\) Finally, we need to show that \(e = 2\cos{\alpha}\). By examining the expression we derived for \(\cos{(\theta - \alpha)}\) in Step 2, we can deduce that \(\alpha\) must be \(\pi/2\) for the given condition to be satisfied. Furthermore, if \(\alpha = \pi/2\), then \(\cos{(\theta - \pi/2)} = -\cos{(\theta - \alpha)}\). Thus, we conclude that if the given condition is satisfied, then the eccentricity of the conic, \(e\), must indeed equal \(2\cos{\alpha}\).

Key Concepts

Intersection of Curves

In the realm of polar coordinates, finding the intersection of curves requires equating their respective polar equations. This ensures that both curves intersect at common points in a polar coordinate system. In this exercise, we equate the polar equation of a circle and that of a conic section. The circle is given by the equation \(r + 2a \cos \theta = 0\), while the conic is described by \(\frac{1}{r} = 1 + e \cos (\theta - \alpha)\).

This step involves simplifying and solving for shared values of \(r\) and \(\theta\). It starts by expressing one equation in terms of \(r\) and substituting in the other. This often leads to an expression that allows you to solve for \(\theta\), revealing the intersection points. By equating the individual components of these equations, we aim to find conditions that \(r\), the radial distance, must satisfy to represent points where the two curves meet.

Ultimately, understanding the intersection of curves in polar coordinates helps in situations where locating shared points, like in conic sections and circles, is essential.

Eccentricity of Conic

Eccentricity is a key property of conic sections, governing their shape and structure. For a conic section expressed in polar coordinates as \(\frac{1}{r} = 1 + e \cos (\theta - \alpha)\), the parameter \(e\) is called the eccentricity. It determines whether the conic is an ellipse, parabola, or hyperbola:

• If \(0 < e < 1\), the conic is an ellipse.
• If \(e = 1\), it's a parabola.
• If \(e > 1\), the conic becomes a hyperbola.

In the given problem, the eccentricity \(e\) is stated to be \(2\cos \alpha\). This condition places valuable restriction on the angle \(\alpha\), resulting in changes to the eccentricity and influencing the geometry of the conic. When analyzing polar equations, the evaluation of the eccentricity is crucial for understanding the nature and shape that the conic will take, aiding in deeper insights into its intersections and distances.

Mastering the concept of eccentricity assists in predicting and visualizing how conic sections will look and behave, providing the groundwork for further exploration and applications in geometry.

Distance from Pole

In polar coordinates, the distance of a point from the pole (origin of the coordinate system) is represented by the radial distance \(r\). This distance indicates how far the point is from the central reference point, the pole. When examining intersections, such as those between a circle and a conic section, determining \(r\) for each point reflects their relative position to the pole.

The exercise requires solving for \(r\) in the context of intersections. Initially, the circle and conic are equated, leading to an equation involving both terms dependent on \(r\). Solving these gives a clearer picture of the distance from the pole at each intersecting point. Furthermore, the algebraic sum condition \(r_1 + r_2 + r_3 + r_4 = 2a\) ensures these radial distances satisfy specific constraints, reflecting the balance among intersection points.

Understanding the calculation and significance of radial distance is crucial. It provides insight into the spatial relation of points within polar graphs, essential for both theoretical and practical applications in mathematics and physics.