Question

Suppose two circles, whose centers are at least \(2 a\) units apart (see figure), are centered at \(F_{1}\) and \(F_{2},\) respectively. The radius of one circle is \(2 a+r\) and the radius of the other circle is \(r,\) where \(r \geq 0 .\) Show that as \(r\) increases, the intersection point \(P\) of the two circles describes one branch of a hyperbola with foci at \(F_{1}\) and \(F_{2}\)

Short answer

Answer: As the radius "r" increases, the intersection point "P" describes one branch of a hyperbola with foci at "F1" and "F2".

Step-by-step solution

Write down the distances from the intersection point to the centers

Let's call the distances between the intersection point \(P\) and the centers of the circles with radii \(2a+r\) and \(r\) as \(PF_1\) and \(PF_2\) respectively.

Create an equation based on the distance definition of a hyperbola

A hyperbola is defined as the set of all points where the difference of the distances from the two foci (in our case \(F_1\) and \(F_2\)) is a constant. Let's call this constant \(2a\), i.e., \(PF_1 - PF_2 = 2a\).

Express the distances in terms of the radii of circles

Using the Pythagorean theorem, we can express the distances \(PF_1\) and \(PF_2\) in terms of the radii of the circles. If the intersection point \(P\) is on both circles, then: \(PF_1 = 2a+r\) \(PF_2 = r\)

Substitute the expressions for the distances into the hyperbola equation

Now, we can substitute the expressions for \(PF_1\) and \(PF_2\) into the equation from Step 2: \((2a+r) - r = 2a\)

Simplify the equation and show that it holds for all values of \(r\)

Simplify the equation: \(2a = 2a\) The equation holds true for all values of \(r\), which means that as \(r\) increases, the intersection point \(P\) always satisfies the definition of a hyperbola with foci at \(F_1\) and \(F_2\). In conclusion, as \(r\) increases, the intersection point \(P\) of the two circles describes one branch of a hyperbola with foci at \(F_1\) and \(F_2\).

Key concepts

Conic Sections

Conic sections—sounds spacey, doesn't it? Well, it is kind of out-of-this-world in a way. Picture this: if you slice through a cone with a plane at different angles and positions, the 2D shapes you get are what we call conic sections. These include circles, ellipses, parabolas, and our star for the day, hyperbolas. They might seem different, but they're like a close-knit family, each shape linked by a set of rules on how they play out in space. Conic sections aren't just a cool math trick; they describe the paths of planets and the beams of lighthouses. So next time you're daydreaming and staring into the sky, remember that conic sections are up there too, carving the paths of celestial bodies.

Locus of Points

Now, don't let the fancy term 'locus' throw you off. A locus is merely a collection of points that share a common property or rule. Imagine having a treasure map with a riddle that guides you to various spots—the positions you end up stepping on form a locus. For a circle, for example, it's all the points that are at an equal distance (the radius) from the center—like a round dancefloor where everyone's an arm's length from the disco ball. Hyperbolas are like that too, but they follow a special rule, which leads us to the heart of their mystery: the distance definition.

Foci of a Hyperbola

Let's zoom in on hyperbolas' secret agents—the foci (pronounced 'foe-sigh'). These aren't just random spots; they're strategic points from which the whole shape of the hyperbola spreads out. You can think of them as two magnetic poles; the entire hyperbola is the field between them. Each point on a hyperbola's curve is on a constant quest to keep the difference in its distance to these foci a steady number. This brings to light the elegant balance of a hyperbola: it's all about maintaining that perfect asymmetry.

Distance Definition of a Hyperbola

Alright, let's get down to the nitty-gritty—what's the rule that defines a hyperbola? It's all about distance. For any point on a hyperbola, the absolute difference in distance to the two foci is always the same. Now, don't mix it up with an ellipse, where the sum of the distances is constant. Here's where our circles from the original exercise come into play: as their intersection point dances farther and closer to each center, it's actually tracing out a hyperbola. It's a graceful ballet that obeys the hyperbola's strict distance rule—an ever-changing radius choreographed by the laws of mathematics.