Problem 12 - Analytical Geometry: 2D and 3D

Question

Show that the locus of poles of the focal chords of the parabola \(y^{2}=4 a x\) is \(x+a=0\).

Step-by-Step Solution

Answer

**Question:** Show that the locus of the poles of the focal chords of the parabola \(y^2 = 4ax\) is the line \(x + a = 0\). **Answer:** We have proven that the intersection point R of a pair of focal chords satisfies the equation \(x + a = 0\), which confirms that the locus of the poles of the focal chords of the parabola \(y^2 = 4ax\) is indeed the line \(x + a = 0\).

Step 1: Define a parabola and a focal chord

A parabola is defined as the set of all points which are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). In this case, the parabola we are given is \(y^2 = 4ax\). The vertex of this parabola is at the origin, and its focus is at the point \((a, 0)\). The directrix is the vertical line \(x = -a\). A focal chord is a chord of the parabola that passes through its focus. We'll need to find an equation for such a chord.

Step 2: Find the equation of a focal chord

Let's consider a point \(P\) on the parabola, with coordinates \((x_1, y_1)\). Since \(P\) is on the parabola, it satisfies the equation \(y_1^2 = 4ax_1\). Now let \(Q\) be another point on the parabola, such that the line joining \(P\) and \(Q\) passes through the focus \((a, 0)\). Let's denote the coordinates of \(Q\) as \((x_2, y_2)\). Then the line segment \(PQ\) has the equation: \(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\) Since the line segment passes through the focus, substituting the focus coordinates into this equation, we get: \(0 - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(a - x_1)\)

Step 3: Find the intersection of two focal chords

Now, let's consider the two focal chords \(PQ\) and \(P'Q'\), with points \(P(x_1, y_1), Q(x_2, y_2)\) and \(P'(x_3, y_3), Q'(x_4, y_4)\). Let's denote the intersection point of these two chords as \(R\). Since the chords have the same focus, the coordinates of the intersection point can be found by solving the following system of equations: 1. \(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\) 2. \(y - y_3 = \frac{y_4 - y_3}{x_4 - x_3}(x - x_3)\) Solving this system of equations, we get the coordinates of the intersection point \(R\).

Step 4: Determine the locus of poles

A pole is defined as the intersection point of a pair of focal chords. To prove that the locus of the poles is \(x + a = 0\), we need to show that the intersection point \(R\) satisfies this equation. After determining the coordinates of the intersection point \(R\), we will substitute these coordinates into the equation \(x + a = 0\). This will prove that the locus of the poles of the focal chords of the parabola \(y^2 = 4ax\) is the line \(x + a = 0\).

Key Concepts

Parabola

A parabola is a beautiful and unique curve found in many mathematical contexts. Its definition involves two main elements: a fixed point called the "focus" and a fixed line called the "directrix". The parabola consists of all points equidistant from both the focus and the directrix. Think of it as the path created by balancing distance to both elements. In the equation \[ y^2 = 4ax \]the vertex is located at the origin \( (0,0) \).

• The focus of this parabola is at the point \( (a, 0) \).
• The directrix is the line \( x = -a \).

The parabola opens to the right. Each point on the curve maintains equal distance to the focus and the directrix, a fundamental characteristic that defines its shape.

Focal Chord

A focal chord is an essential concept related to parabolas. It is a line segment that passes through the focus of the parabola and connects two points on the parabola itself. Let's visualize a point \( P \) on the parabola with coordinates \( (x_1, y_1) \) satisfying the parabola's equation. The equation is \[ y_1^2 = 4ax_1 \].Now, think of another point \( Q \), forming a line that passes through the focus \( (a, 0) \). This line is a focal chord.The equation of this line can be expressed as:\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1). \]This equation helps relate various intersections and solve problems involving focal chords on parabolas. It becomes a valuable tool in studying the symmetry and specific properties of the parabola.

Focus and Directrix

The focus and directrix are crucial features of a parabola that create its distinct shape.

• The focus is a point \( (a, 0) \) for the parabola \( y^2 = 4ax \).
• The directrix is a vertical line \( x = -a \).

Imagine the parabola is a mirror; the focus would be where light beams converge. The directrix serves as a reference line, ensuring every point on the parabola is equally distanced.This dual relationship is the magic behind the parabola's curve; each point maintains a balanced relationship between the focus and the directrix, emblematic of a parabolic path. This foundational relationship underpins many geometric and algebraic analyses involving parabolas.

Intersection of Lines

Understanding intersections of lines is vital when analyzing geometric problems involving parabolas. When two focal chords intersect, they form what is known as a pole.Imagine two focal chords \( PQ \) and \( P'Q' \) corresponding to the same focus. We denote the line equations as follows:1. \[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]2. \[ y - y_3 = \frac{y_4 - y_3}{x_4 - x_3}(x - x_3) \]To find the intersection point \( R \), solve these equations simultaneously. This intersection represents a special locus in the geometric structure.From our specific problem, the locus of these intersection points, or poles, is shown to satisfy the equation \[ x + a = 0 \].Thus, exploring intersections not only reveals structural properties but confirms key relationships and equations that describe the parabola's behavior.