Problem 14 - Analytical Geometry: 2D and 3D

Question

Prove that the polar of any point on the circle \(x^{2}+y^{2}-2 a x-3 a^{2}=0\) with respect to the circle \(x^{2}+y^{2}+2 a x-3 a^{2}=0\) touches the parabola \(y^{2}=4 a x\).

Step-by-Step Solution

Answer

Short answer: To prove that the polar of any given point on a given circle with respect to another circle touches a specific parabola, we first found the polar equation of the given circle and the parabola. By showing the discriminant of the quadratic in y is zero, we concluded that the polar line touches the given parabola.

Step 1: Find the polar equation of the given circles

Let the circles be \(C_1: x^2 + y^2 - 2ax - 3a^2 = 0\) and \(C_2: x^2 + y^2 + 2ax - 3a^2 = 0\). Let's rewrite them in the form \((x-a)^2 + y^2 = r^2\) to find their centers and radii. Center of \(C_1\) is \((a, 0)\) with radius \(r_1 = 2a\) Center of \(C_2\) is \((-a, 0)\) with radius \(r_2 = 2a\)

Step 2: Find the equation of the polar line

Let \(P(u, w)\) be a point on circle \(C_1\). The equation of the polar with respect to circle \(C_2\) can be obtained from the circle's equation by replacing \(x\), \(y\), and the constant term with \(ux + wy\), \(wx - uy\), and \(u(-a) + w(0) - 3a^{2}\). Therefore, the polar equation is: \((u(x + a) + w(y))^2 = r_2^2 (u^2 + w^2)\)

Step 3: Show that the polar touches the parabola

The parabola equation is given by \(y^2 = 4ax\). First, let's substitute the equation in the polar equation above: \((u(x + a) + w(4ax - y))^2 = r_2^2 (u^12 + w^2)\) Next, we need to show that the polar equation touches the parabola. For this, the discriminant of the quadratic in \(y\) must be zero. Let's find it and set it equal to zero: Discriminant = \(B^2 - 4AC\), where \(A = w^2 - 1\), \(B = -8aw\), and \(C = u^2 (x^2 - 4ax - a^2)\). Setting the discriminant equal to zero and simplifying, we get: \(64a^2 w^2 - 4(w^2 - 1)(u^2 (x^2 - 4ax - a^2)) = 0\) Divide both sides by \(4(w^2 - 1)\), we have: \(16 a^2 w^2 = u^2 (x^2 - 4ax - a^2)\) Since the polar line of point \(P\) with respect to circle \(C_2\) has this discriminant to be zero, we can conclude that it touches the parabola \(y^2 = 4ax\).

Key Concepts

Circles in Polar Coordinates

Understanding circles in polar coordinates can make exploring geometric relationships more intuitive. A circle can be represented using a center point and a radius. In polar coordinates, a circle's equation is usually given as \(r = a \, \cos \theta + b \, \sin \theta\). This links to how coordinates are described in this system.

In the exercise, we dealt with two circles, \(C_1\) and \(C_2\), with centers at \( (a, 0) \) and \((-a, 0)\) respectively in Cartesian coordinates. This highlights how the center of a circle in polar coordinates shifts based on orientation and location. The transformation between Cartesian and polar coordinates involves identifying these shifts and representing the circle's form accordingly.

When analyzing circles in polar coordinates, always concentrate on the center location and the radius as these define the circle's position and size.

Parabola

A parabola is a symmetrical, U-shaped curve, described mathematically by the quadratic equation \(y^2 = 4ax\). Understanding the role of vertices and axes is key when working with parabolas. The vertex represents the peak or lowest point, depending on the direction the parabola opens.

In the problem, the parabola \(y^2 = 4ax\) plays a crucial role. This parabola represents a curve that all polar lines derived from points on circle \(C_1\) touch. Recognizing how the vertex and focus of the parabola relate to its equation helps understand the conditions of touch—a tangent condition. This involves linking the geometric place (parabola) with the line's trajectory (polar line).

The parabola is central in this problem, serving as the curve which the polar lines will tangentially touch.

Discriminant in Quadratic Equations

The discriminant of a quadratic equation, expressed as \(B^2 - 4AC\), offers insight into the nature of a quadratic curve's solutions. It tells us whether the solutions are real and distinct, real and repeated, or complex. This is pivotal in demonstrating tangency in this context.

In the problem, we needed to show that the polar line touches the parabola. By examining the quadratic form produced when substituting into the parabola's equation, setting the discriminant equal to zero confirms tangency. This indicates that at least one solution of the quadratic equation \(A = w^2 - 1, B = -8aw, C= u^2(x^2 - 4ax - a^2)\) is repeated, implying a single touching point.

Understanding how the discriminant affects the behavior of quadratic equations builds a bridge from algebra to geometry in polar contexts.

Polar Equations

Polar equations describe a relationship between the radial distance \(r\) and the angle \(\theta\). They are a powerful way to depict curves not easily described in Cartesian form. These equations are particularly useful in depicting curves and understanding geometric transformations.

In the exercise, the polar of a point on a circle turns out to be a line which must tangentially intersect a parabola. When we derive the polar line with respect to circle \(C_2\), we replace Cartesian components with polar equivalents. The complexity of this substitution, all coming back to the core form \(r = a \, \cos \theta + b \, \sin \theta\), helps identify how the tangent behavior emerges.

Not only do polar equations simplify the structure of certain curves, but they also serve as key tools in proving and understanding complex geometric properties, such as tangency and intersection.