Photo AI
The 3D graphic below shows a pencil in the form of a regular prism - Leaving Cert DCG - Question A-3 - 2016
Question A-3
The 3D graphic below shows a pencil in the form of a regular prism. When a conical top is applied to the pencil by the sharpener it results in a series of hyperbolic... show full transcript
Worked Solution & Example Answer:The 3D graphic below shows a pencil in the form of a regular prism - Leaving Cert DCG - Question A-3 - 2016
Step 1
Locate the position of the vertex and draw a portion of the hyperbola.
Answer
The vertex of the hyperbola can be found at the intersection of the axes AA1 and DD1. Given the hyperbola's parameters, the vertex lies at the midpoint between the focus and directrix.
To draw a portion of the hyperbola, start by plotting the vertex point. Then, using the eccentricity (3.2), determine the relationship between the distance from the focus to the directrix. Using this information, sketch one branch of the hyperbola with the correct curvature.
Step 2
Locate points on and outside latus rectum.
Answer
The latus rectum can be calculated using the formula derived from the hyperbola's equation and eccentricity. For a hyperbola, the latus rectum is given by: Here, 'a' and 'b' can be determined from eccentricity and the conic's directrix. Points on the latus rectum can be plotted above and below the focal point. Identify these key points as they assist in constructing the curve.
Step 3
Locate point inside latus rectum.
Answer
Choose a point that lies within the bounds created by the latus rectum. This point will help refine the overall shape of the hyperbolic curve. Using the geometric properties of hyperbolas, such a point can usually be selected to maintain proper proportional distances.
Step 4
Draw curve (Any = 1).
Answer
Using the points gathered (the vertex, points on and outside the latus rectum, and the inner point), sketch the hyperbola. Start at the vertex and smoothly transition through the selected points ensuring hyperbolic characteristics are maintained. Emphasize that only one branch needs to be drawn as per the instructions.
Step 5
Step 6
Construct tangent to the curve at point P.
Answer
To construct a tangent at the point P, first draw a line perpendicular to the radius drawn from the focus to point P. Utilize the properties of hyperbolas to ensure the tangent line accurately reflects the curve's slope at that point. The tangent line should touch the hyperbola only at point P.