Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 31-34: Find an equation for the set of points in an \(x y\)-plane that are equidistant from the point \(P\) and the line \(l\). $$ P(7,0) ; \quad l: x=1 $$

Step-by-Step Solution

Answer

The set of points form a parabola, and its equation is \(y^2 = 12x - 48\).

Step 1: Understanding the Problem

To find the equation of the set of points equidistant from a point \(P\) and a line \(l\), we need to apply the definition of a parabola. A point is on the parabola if it is equidistant from a fixed point \(P\), called the focus, and a line \(l\), called the directrix.

Step 2: Express the Distance to Point P

The focus of the parabola is given by the point \(P(7,0)\). The distance from any point \((x, y)\) to the focus \(P(7,0)\) is given by the formula: \ \[ d_1 = \sqrt{(x - 7)^2 + y^2} \]

Step 3: Express the Distance to Line l

The directrix of the parabola is the line \(x = 1\). The distance from any point \((x, y)\) to the line \(x = 1\) is the horizontal distance, calculated as: \ \[ d_2 = |x - 1| \]

Step 4: Setting Up the Equation

Since the point \((x, y)\) is equidistant from both the point and the line, we equate the two distances: \ \[ \sqrt{(x - 7)^2 + y^2} = |x - 1| \]

Step 5: Squaring Both Sides

To eliminate the square root, square both sides of the equation. This yields: \ \[ (x - 7)^2 + y^2 = (x - 1)^2 \]

Step 6: Expanding and Simplifying

Expand both sides of the equation: \ Left Side: \ \( (x-7)^2 + y^2 = x^2 - 14x + 49 + y^2 \) \ Right Side: \ \( (x-1)^2 = x^2 - 2x + 1 \). \ After simplifying: \ \( x^2 - 14x + 49 + y^2 = x^2 - 2x + 1 \) \ Remove \(x^2\) from both sides: \ \(-14x + 49 + y^2 = -2x + 1\)

Step 7: Isolating y^2

To solve for \(y^2\), isolate it on one side of the equation: \ \[ y^2 = 12x - 48 \] This is the equation of the parabola in its simplest form.

Key Concepts

Distance Formula

The distance formula is a mathematical equation used to determine the distance between two points in a Cartesian coordinate system. It's a handy tool, especially when dealing with problems involving geometry on a plane. The formula itself comes from the Pythagorean theorem and is expressed as:

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This formula calculates the distance \(d\) between two points, \((x_1, y_1)\) and \((x_2, y_2)\). In the context of our problem, it is used to find the distance from a point \((x, y)\) to the focus \((7,0)\), simplifying to:

• \[ d_1 = \sqrt{(x - 7)^2 + y^2} \]

This gives us a way to express the distance in terms of \(x\) and \(y\), which we will set equal to the distance calculated from the line in our later steps.

Focus and Directrix

A parabola is a symmetric curve that can be defined with the help of a fixed point and a fixed line. This fixed point is known as the focus, and the fixed line is called the directrix. The fascinating thing about a parabola is its defining property: every point on the parabola is equidistant from its focus and directrix.
In the problem provided, we have:

• Focus: The point \(P(7,0)\)
• Directrix: The line \(x=1\)

To understand how this property sets a parabola, imagine placing a point \((x, y)\) so its distance to the focus \(P(7,0)\) equals its distance to the line \(x=1\). By considering these distances, we generate the distinctive U-shaped curve of a parabola. By using algebraic expressions for these distances and equating them, we derive a parabolic equation such as \(y^2 = 12x - 48\), which represents the set of all such equidistant points.

Equidistant Points

Equidistant points in a geometrical setting mean points which maintain the same distance from certain objects, such as lines or other points. For parabolas, this concept plays a central role. Here, it means each point on the parabola maintains equal distance from the focus \(P(7,0)\) and the directrix \(x=1\).
To visualize this, imagine drawing a string tight between the focus and the directrix, and letting a pencil trace a path while maintaining the string taut. Each position the pencil adopts under this restriction is at an equal distance from both the focus and directrix, painting the familiar parabola curve.
By equating those distances using their algebraic expressions, we simplify them to derive the equation representing all such equidistant points. This results in the parabolic equation \((x - 7)^2 + y^2 = (x - 1)^2\), which gives us a clear, specific form of the radial **parabolic relationship** between the points on the curve and their defining elements, the focus and directrix.