The Locus Definition Of A Parabola Homework Answers

The parabola, a fundamental shape in mathematics and physics, isn't just a curve you see on graphs or in projectile motion. It's a geometric marvel defined by a very specific rule: its locus definition. Understanding this definition unlocks a deeper understanding of the parabola and its properties. This article will explore the locus definition of a parabola, providing homework-style answers and explanations to solidify your grasp of the concept.

Delving into the Locus Definition: What is a Parabola, Really?

At its core, a parabola is the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).

• Focus: A specific point inside the curve of the parabola.
• Directrix: A specific line outside the curve of the parabola.
• Equidistant: Meaning "the same distance." For any point on the parabola, the distance to the focus is identical to the distance to the directrix.

Imagine a point 'P' moving in such a way that its distance to a single, stationary point (the focus 'F') is always equal to its distance to a straight line (the directrix 'D'). The path that point 'P' traces out is a parabola.

This definition is crucial because it doesn't rely on equations like y = x². Instead, it provides a geometric foundation for understanding what a parabola is, regardless of its orientation or position in a coordinate plane.

Deconstructing the Definition: Key Concepts and Terms

Before tackling homework problems, let's clarify some essential elements:

• Vertex: The point on the parabola that is closest to both the focus and the directrix. It lies exactly halfway between the focus and the directrix. The vertex is the "turning point" of the parabola.
• Axis of Symmetry: A line that passes through the focus and is perpendicular to the directrix. It divides the parabola into two symmetrical halves. The vertex lies on the axis of symmetry.
• Focal Length (p): The distance from the vertex to the focus (or, equivalently, the distance from the vertex to the directrix).
• Latus Rectum: A line segment passing through the focus, perpendicular to the axis of symmetry, with endpoints on the parabola. Its length is equal to 4p (four times the focal length). The latus rectum helps define the "width" of the parabola.

Homework Answers and Explanations: Putting the Locus Definition to Work

Now, let's apply the locus definition to solve some typical parabola homework problems.

Problem 1: Finding the Equation of a Parabola Given its Focus and Directrix

Question: A parabola has a focus at (2, 3) and a directrix of y = -1. Find the equation of the parabola.

Answer and Explanation:

1. Apply the Locus Definition: Let (x, y) be any point on the parabola. According to the locus definition, the distance from (x, y) to the focus (2, 3) must equal the distance from (x, y) to the directrix y = -1.

2. Calculate the Distance to the Focus: We use the distance formula:

   Distance to Focus = √[(x - 2)² + (y - 3)²]

3. Calculate the Distance to the Directrix: The distance from a point (x, y) to a horizontal line y = c is simply the absolute value of the difference in their y-coordinates:

   Distance to Directrix = |y - (-1)| = |y + 1|

4. Set the Distances Equal: Now, we equate the two distances, reflecting the locus definition:

   √[(x - 2)² + (y - 3)²] = |y + 1|

5. Solve for the Equation: This involves some algebra:

   • Square both sides to eliminate the square root: (x - 2)² + (y - 3)² = (y + 1)²
   • Expand the terms: x² - 4x + 4 + y² - 6y + 9 = y² + 2y + 1
   • Simplify by cancelling the y² terms and rearranging: x² - 4x + 13 - 6y = 2y + 1
   • Isolate the y terms: x² - 4x + 12 = 8y
   • Solve for y: y = (1/8)x² - (1/2)x + (3/2)

Therefore, the equation of the parabola is y = (1/8)x² - (1/2)x + (3/2).

Problem 2: Determining the Focus and Directrix from the Equation of a Parabola

Question: The equation of a parabola is x² = 12y. Find the focus and directrix of the parabola.

Answer and Explanation:

1. Recognize the Standard Form: The equation x² = 12y is a standard form parabola opening upwards. The general form for such a parabola is x² = 4py, where p is the focal length.

2. Find the Focal Length (p): Comparing x² = 12y to x² = 4py, we see that 4p = 12. Therefore, p = 3.

3. Determine the Focus: Since the parabola opens upwards and the vertex is at the origin (0, 0), the focus is located p units above the vertex. Thus, the focus is at (0, 3).

4. Determine the Directrix: The directrix is a horizontal line located p units below the vertex. Therefore, the directrix is the line y = -3.

Problem 3: Finding the Vertex, Focus, and Directrix Given a General Form Equation

Question: Find the vertex, focus, and directrix of the parabola defined by the equation y² - 4y - 8x + 20 = 0.

Answer and Explanation:

1. Rewrite the Equation in Standard Form: We need to complete the square to get the equation into a more recognizable form. Since the y term is squared, we'll complete the square with respect to y.

   • Rearrange the equation: y² - 4y = 8x - 20
   • Complete the square on the left side: Take half of the coefficient of the y term (-4), square it ( (-2)² = 4), and add it to both sides: y² - 4y + 4 = 8x - 20 + 4
   • Rewrite the left side as a squared term: (y - 2)² = 8x - 16
   • Factor out the coefficient of x on the right side: (y - 2)² = 8(x - 2)

2. Identify the Vertex: The standard form of a parabola opening to the right is (y - k)² = 4p(x - h), where (h, k) is the vertex. Comparing this to our equation, we see that the vertex is (2, 2).

3. Find the Focal Length (p): From the equation (y - 2)² = 8(x - 2), we have 4p = 8, so p = 2.

4. Determine the Focus: Since the parabola opens to the right, the focus is located p units to the right of the vertex. Therefore, the focus is at (2 + 2, 2) = (4, 2).

5. Determine the Directrix: The directrix is a vertical line located p units to the left of the vertex. Therefore, the directrix is the line x = 2 - 2 = 0, or simply x = 0.

Problem 4: Applying the Locus Definition to a Geometric Problem

Question: Point P moves in the xy-plane such that its distance from point A(1,1) is always equal to its distance from the line y = -x. Find the equation of the locus of point P.

Answer and Explanation:

1. Define Point P: Let P have coordinates (x, y).

2. Distance to Point A (Focus): Using the distance formula, the distance from P(x, y) to A(1, 1) is:

   Distance to A = √[(x - 1)² + (y - 1)²]

3. Distance to the Line y = -x (Directrix): The distance from a point (x, y) to a line Ax + By + C = 0 is given by the formula:

   Distance = |Ax + By + C| / √(A² + B²)

   In our case, the line y = -x can be rewritten as x + y = 0. Therefore, A = 1, B = 1, and C = 0. The distance from (x, y) to the line x + y = 0 is:

   Distance to Line = |x + y| / √(1² + 1²) = |x + y| / √2

4. Apply the Locus Definition: Equate the two distances:

   √[(x - 1)² + (y - 1)²] = |x + y| / √2

5. Solve for the Equation:

   • Square both sides: (x - 1)² + (y - 1)² = (x + y)² / 2
   • Expand: x² - 2x + 1 + y² - 2y + 1 = (x² + 2xy + y²) / 2
   • Multiply both sides by 2: 2x² - 4x + 2 + 2y² - 4y + 2 = x² + 2xy + y²
   • Simplify and rearrange: x² - 2xy + y² - 4x - 4y + 4 = 0

Therefore, the equation of the locus of point P is x² - 2xy + y² - 4x - 4y + 4 = 0. This equation represents a parabola, but it's rotated compared to the standard forms we typically encounter.

Problem 5: Using the Locus Definition to Construct a Parabola

Question: Given a focus F and a directrix D, describe how to construct a parabola using only a compass and straightedge, based on the locus definition.

Answer and Explanation:

1. Draw the Focus and Directrix: Begin by drawing a point F (the focus) and a line D (the directrix) on your paper. Ensure that the point is not on the line.

2. Draw Lines Perpendicular to the Directrix: Draw several lines perpendicular to the directrix D. These lines will be parallel to the axis of symmetry.

3. Find Points on the Parabola: For each line perpendicular to the directrix:

   • Measure the distance from the perpendicular line to the directrix. Let's call this distance 'd'.
   • Using a compass, set its radius to 'd'.
   • Place the compass point on the focus F.
   • Draw arcs that intersect the perpendicular line. The points where the arcs intersect the perpendicular line are points on the parabola. This is because these points are, by construction, equidistant from the focus (F) and the directrix (D).

4. Connect the Points: Carefully connect the points you've found with a smooth curve. This curve is the parabola.

5. Repeat: Repeat step 3 and 4 for more perpendicular lines to create a more accurate and complete parabola. The more lines you use, the smoother and more accurate your parabola will be.

Deeper Understanding: The Parabola in Context

The locus definition isn't just a theoretical exercise. It provides a fundamental understanding of why parabolas appear in various contexts:

• Optics: Parabolic mirrors and reflectors are designed based on the locus definition. Light rays (or other electromagnetic waves) emanating from the focus will be reflected off the parabolic surface in parallel lines. Conversely, parallel rays entering the parabola will be focused onto the focus. This principle is used in telescopes, satellite dishes, and car headlights.

• Projectile Motion: Ignoring air resistance, the path of a projectile (like a ball thrown through the air) is a parabola. This is because gravity provides a constant downward acceleration, which can be modeled mathematically to create a parabolic trajectory.

• Engineering: Parabolic arches are strong and efficient structures because they distribute weight evenly.

Common Mistakes to Avoid

• Confusing Focus and Directrix: Remember the focus is a point inside the curve, while the directrix is a line outside the curve.
• Incorrect Distance Calculation: Ensure you're using the correct distance formula for point-to-point and point-to-line distances.
• Algebra Errors: Be careful when squaring and simplifying equations. A small error can lead to an incorrect equation for the parabola.
• Forgetting the Absolute Value: When calculating the distance to the directrix, remember to use the absolute value to ensure the distance is always positive.

FAQs about the Locus Definition of a Parabola

Q: Why is the locus definition important?

A: It provides a fundamental geometric understanding of what a parabola is, independent of any coordinate system or algebraic equation. It highlights the defining property of equal distances.

Q: Can any point and line be used to define a parabola?

A: No. The point (focus) cannot lie on the line (directrix). If the point were on the line, the locus of points equidistant from both would be a straight line (the perpendicular bisector of the segment connecting the point to the line).

Q: How does the focal length affect the shape of the parabola?

A: A larger focal length results in a "wider" or "flatter" parabola. A smaller focal length results in a "narrower" or "steeper" parabola.

Q: Is there a locus definition for other conic sections (ellipse, hyperbola)?

A: Yes! Ellipses and hyperbolas can also be defined by loci, but they involve the ratio of distances to a focus and a directrix. For an ellipse, this ratio (called the eccentricity) is less than 1. For a hyperbola, it's greater than 1. For a parabola, the eccentricity is exactly 1.

Conclusion: Mastering the Parabola Through its Definition

Understanding the locus definition of a parabola is more than just memorizing a definition. It's about grasping the underlying geometric principle that governs the shape of this important curve. By working through problems, visualizing the definition, and understanding its applications, you can achieve a deeper and more meaningful understanding of parabolas and their role in mathematics and the real world. Practice applying the locus definition, and you'll find that parabolas become much less mysterious and much more intuitive.