Question

Find the equation for the set of points the difference of whose distances from \((5,0)\) and \((-5,0)\) is 6 units.

Short answer

The equation representing the set of points \((x,y)\) whose difference between their distances from \((5,0)\) and \((-5,0)\) is 6 units is \(4x^2=(x\pm5)^2+y^2\).

Step-by-step solution

Set up the equation

We are given the condition: \[|\sqrt{(x-5)^2+y^2}-\sqrt{(x+5)^2+y^2}|=6\]

Remove the absolute value

Since the difference between the two distances can be either positive or negative, we will consider two cases: Case 1: \[\sqrt{(x-5)^2+y^2}-\sqrt{(x+5)^2+y^2}=6\] Case 2: \[\sqrt{(x+5)^2+y^2}-\sqrt{(x-5)^2+y^2}=6\]

Solve each case

Let's solve each case separately: Case 1: 1. Add the second square root to both sides: \[\sqrt{(x-5)^2+y^2}=\sqrt{(x+5)^2+y^2}+6\] 2. Square both sides: \[(x-5)^2+y^2=((x+5)^2+y^2)+12\sqrt{(x+5)^2+y^2}+36\] 3. Simplify and cancel terms: \[24x=12\sqrt{(x+5)^2+y^2}\] 4. Squaring both sides again: \[576x^2=144((x+5)^2+y^2)\] 5. Divide both sides by 144: \[4x^2=(x+5)^2+y^2\] Case 2: 1. Add the first square root to both sides: \[\sqrt{(x+5)^2+y^2}=\sqrt{(x-5)^2+y^2}+6\] 2. Square both sides: \[(x+5)^2+y^2=((x-5)^2+y^2)+12\sqrt{(x-5)^2+y^2}+36\] 3. Simplify and cancel terms: \[-24x=12\sqrt{(x-5)^2+y^2}\] 4. Squaring both sides again: \[576x^2=144((x-5)^2+y^2)\] 5. Divide both sides by 144: \[4x^2=(x-5)^2+y^2\]

Combine the equations

Both cases lead to the same equation: \[4x^2=(x\pm5)^2+y^2\] So, this is the equation that represents the set of points \((x,y)\) whose difference between their distances from \((5,0)\) and \((-5,0)\) is 6 units.

Key concepts

Distance Formula

Understanding the distance formula is crucial for solving problems involving distances between points in a coordinate plane. The distance formula is derived from the Pythagorean theorem, and it calculates the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) as \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). The formula essentially measures the hypotenuse of a right-angled triangle formed by these points.

Let's apply this to find the difference in distances from a point \( (x, y) \) to two fixed points. If we take \( (5, 0) \) and \( (-5, 0) \) as our fixed points, the distances from \( (x, y) \) to these points are determined by plugging these coordinates into the distance formula, resulting in \( \sqrt{(x-5)^2 + y^2} \) and \( \sqrt{(x+5)^2 + y^2} \) respectively.

Absolute Value in Equations

The absolute value measures the magnitude of a number regardless of its sign. It's always non-negative, as it gives the distance of the number from zero on the real number line. When we encounter absolute values in equations, it indicates that there are two scenarios: one where the quantity inside the absolute value is positive, and one where it is negative.

In our exercise, we are dealing with an equation representing the absolute difference in distance, written as \(|\sqrt{(x-5)^2+y^2}-\sqrt{(x+5)^2+y^2}|=6\). This leads us to consider two separate cases for the equation based on the properties of absolute values: one where the first distance is greater than the second \( \sqrt{(x-5)^2+y^2} - \sqrt{(x+5)^2+y^2} = 6 \) and another where it is less \( \sqrt{(x+5)^2+y^2} - \sqrt{(x-5)^2+y^2} = 6 \)—each of which is solved to find the equation of the set of points in question.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The four basic types of conic sections are circles, ellipses, parabolas, and hyperbolas. Each of these has a unique equation that models its shape. For instance, circles are represented by \((x-h)^2 + (y-k)^2 = r^2\), while ellipses and hyperbolas have more complex equations involving the squares of \(x\) and \(y\) terms.

In the context of our exercise, the final equation obtained, \(4x^2 = (x\pm5)^2 + y^2\), resembles that of a conic section. Depending on further manipulation and the completion of the square, this equation could represent a hyperbola, as it involves the subtraction of distances, which is characteristic of hyperbolas. Conic sections are widely used in many fields including astronomy, physics, engineering, and art.