Locus Of A Point: Eccentricity 3/4 & Curve ID

by TextBrain Team 46 views

Alright, guys, let's dive into a fascinating problem involving the locus of a point, eccentricity, and identifying a curve. This is a classic problem in geometry that combines understanding of conic sections with the practical skill of drawing. So, grab your pencils, compasses, and let’s get started!

Understanding the Problem

First, let’s break down what we need to do. We're asked to find the path (or locus) traced by a point P. This point moves according to a specific rule: its eccentricity is always ¾. In simpler terms, eccentricity is a measure of how much a conic section (like an ellipse, parabola, or hyperbola) deviates from being a perfect circle. It’s the ratio of the distance from a point on the curve to a fixed point (called the focus) and the distance from that same point to a fixed line (called the directrix).

We're given that the fixed point (focus) is 50 mm away from the fixed straight line (directrix). This is crucial information because it sets the scale for our drawing. The goal is to accurately plot the path of point P and then identify what kind of curve we've created. Is it an ellipse? A parabola? Or a hyperbola? Given the eccentricity of ¾, we already have a strong hint, but let's confirm it visually and mathematically.

So, to recap, we need to:

  1. Draw the directrix and mark the focus.
  2. Use the eccentricity ratio to find points on the locus.
  3. Connect the points to form the curve.
  4. Identify the curve based on its shape and the given eccentricity.

Setting Up the Drawing

Before we start plotting points, we need to set up our drawing accurately. This initial setup is super important, as any errors here will propagate through the rest of the construction.

  1. Draw the Directrix: On your drawing sheet, use a ruler to draw a straight line. This will be our directrix. Make sure it's long enough to accommodate the curve we're about to draw. A good length would be at least 150 mm.
  2. Mark the Axis: Draw a line perpendicular to the directrix. This line is the axis of our conic section. It passes through the focus and is perpendicular to the directrix. The intersection of the axis and directrix will be our reference point.
  3. Locate the Focus: Measure 50 mm along the axis from the directrix. Mark this point clearly. This is our focus (usually labeled as F). Double-check this measurement to ensure accuracy.

With these three elements – the directrix, the axis, and the focus – we have the foundation for constructing the locus of point P. Make sure everything is clear and well-defined before moving on to the next step.

Plotting Points on the Locus

Now comes the exciting part – finding the points that satisfy the eccentricity condition. Remember, the eccentricity (e) is defined as:

e = PF / PD

Where:

  • PF is the distance from point P to the focus F.
  • PD is the distance from point P to the directrix.

In our case, e = ¾. This means PF = (¾) * PD. We'll use this relationship to find various points on the locus.

Here’s how we do it:

  1. Choose a Distance from the Directrix: Pick a distance along the axis from the directrix. Let’s say we choose 20 mm. This will be our PD.
  2. Calculate the Corresponding Distance from the Focus: Using the eccentricity ratio, calculate PF. PF = (¾) * 20 mm = 15 mm.
  3. Draw an Arc: Using the focus F as the center, draw an arc with a radius of 15 mm. This arc represents all possible locations of point P that are 15 mm away from the focus.
  4. Draw a Line Parallel to the Directrix: At a distance of 20 mm from the directrix (on the same side as the focus), draw a line parallel to the directrix. This line represents all possible locations of point P that are 20 mm away from the directrix.
  5. Find the Intersection: The point where the arc and the line intersect is a point on the locus of P. This is because at this point, PF = 15 mm and PD = 20 mm, satisfying our eccentricity condition.
  6. Repeat: Repeat this process with different distances from the directrix (e.g., 30 mm, 40 mm, 50 mm, etc.). Each time, calculate the corresponding distance from the focus and find the intersection. The more points you plot, the more accurate your curve will be.

Remember to plot points on both sides of the axis to get a symmetrical curve. Aim for at least 10-12 points to get a good representation of the locus.

Sketching the Curve

Once you have a good number of points plotted, carefully sketch a smooth curve through them. The curve should pass through all the points, and it should be symmetrical about the axis. Use a flexible curve or French curve to help you draw a smooth line. If you don’t have those tools, just do your best to connect the points freehand.

As you sketch the curve, observe its shape. Does it look like a circle, an ellipse, a parabola, or a hyperbola? Remember, we know the eccentricity is ¾, which is less than 1. This should give you a strong clue about the type of curve you're drawing.

Identifying the Curve

Based on the eccentricity and the shape of the curve, we can now identify the curve. Since the eccentricity e = ¾, which is less than 1 (0 < e < 1), the curve is an ellipse.

An ellipse is a conic section formed by the intersection of a plane and a cone when the angle between the plane and the axis of the cone is greater than the angle between the generator of the cone and its axis. In simpler terms, it’s like a stretched circle. The closer the eccentricity is to 0, the more the ellipse resembles a circle. As the eccentricity approaches 1, the ellipse becomes more elongated.

So, the final answer is: The curve is an ellipse.

Key Takeaways

  • Eccentricity is a key parameter that defines the shape of conic sections.
  • An eccentricity between 0 and 1 indicates an ellipse.
  • Accurate setup and plotting of points are crucial for drawing the locus accurately.
  • The formula e = PF/PD is fundamental to solving locus problems involving eccentricity.

Additional Tips for Accuracy

  • Use a sharp pencil to draw precise lines and arcs.
  • Double-check all measurements to avoid errors.
  • Plot enough points to get a clear shape of the curve.
  • Use a flexible curve or French curve to draw smooth lines.
  • Practice makes perfect! The more you practice these types of problems, the better you’ll become at visualizing and drawing conic sections.

Conclusion

Drawing the locus of a point based on its eccentricity is a fundamental problem in geometry that combines theoretical understanding with practical drawing skills. By carefully setting up the drawing, plotting points according to the eccentricity ratio, and sketching the curve, we were able to identify the curve as an ellipse. Remember the key concepts, practice regularly, and you'll master these types of problems in no time! Keep exploring and happy drawing, everyone!