The graphic below shows the trophy for the FIFA Club World Cup - Leaving Cert DCG - Question A-2 - 2015
Question A-2
The graphic below shows the trophy for the FIFA Club World Cup. The small outline elevation, which is also given, shows that the trophy is based on two identical par... show full transcript
Worked Solution & Example Answer:The graphic below shows the trophy for the FIFA Club World Cup - Leaving Cert DCG - Question A-2 - 2015
Step 1
Establish points on one semi parabola …(Min 5 incl. vertex and end point)
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Answer
To find the points on one side of the parabola, we first establish the vertex at V₁ and an endpoint at A. The standard equation for a parabola can be used:
y=a(x−h)2+k
where (h, k) is the vertex. Choose suitable values of 'a' to define the curvature. Calculate y for various x values to find additional points.
Step 2
Establish points on other half of parabola ...(mirror or otherwise)
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Using the symmetry of the parabola, the corresponding points on the opposite side can be determined. If (x, y) are the coordinates on one side, then the equivalent points on the other side will be (-x, y). This maintains symmetry about the vertex.
Step 3
Establish points on 2nd parabola ...(mirror or otherwise)
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Repeat the process for the second parabola (defined by points C and D). Again, use the standard form for a parabola and find points similarly to ensure they mirror the first parabola.
Step 4
Determination of positions for points B, D, P & Q
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Points B, D, P, and Q can be determined based on their distances from the vertex and focal points. For instance, if P lies closer to the focus F₁ and D is at the midpoint of the directrix, their coordinates can be calculated based on the general parabola equations.
Step 5
Draw parabolic curves …
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Answer
Using the points established, sketch the parabolas accurately on the provided diagram. Ensure that they reflect the correct shapes and intersect at the right coordinates.
Step 6
Draw lines from P (or Q) to foci …
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From point P, draw lines to each of the foci F₁ and F₂. This visually represents the relationship between the focal points and the parabola, confirming that points on the curve are equidistant from the focus and the directrix.
Step 7
Bisect angle and draw normal through P (or Q) …
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To find the normal at point P, determine the angle between the lines drawn from P to the foci and bisect that angle. The line perpendicular to the tangent at P (the normal) can then be drawn.
Step 8
Establish centre and draw required circle
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The center of the circle tangential to the parabolas can be found at the midpoint of line BQ or DP, depending on which side is used. From this center, draw a circle that touches points P and Q, ensuring it is tangent to the curves.
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