The image below shows a parabolic bridge at the M50 interchange in Dublin - Leaving Cert DCG - Question A-3 - 2020
Question A-3
The image below shows a parabolic bridge at the M50 interchange in Dublin. The drawing shows the directrix DD, the focus F and two points P and Q on a similar parabo... show full transcript
Worked Solution & Example Answer:The image below shows a parabolic bridge at the M50 interchange in Dublin - Leaving Cert DCG - Question A-3 - 2020
Step 1
Locate the axis
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Answer
The axis of the parabola is the vertical line that runs through the focus F, which is the midpoint between the directrix DD and the vertex. This is the line of symmetry for the parabola.
Step 2
Location of vertex
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The vertex is located at the point where the parabola changes direction. It lies on the axis of symmetry, equidistant from the directrix and the focus. Calculate its position based on these distances.
Step 3
Locate points below latus rectum
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Using the distance between the focus and the vertex, locate the points that intersect the latus rectum. These points lie horizontally from the focus and below the vertex.
Step 4
Locate points above latus rectum
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Similar to the previous step, locate points that are directly vertically above the latus rectum, ensuring they align with the parabola's curve.
Step 5
Draw curve
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Sketch the parabolic curve starting from point P and ensuring it passes through Q, maintaining symmetry about the axis and confirming that the curve smoothly transitions through all points identified.
Step 6
Construct a normal to the curve at the point P
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To construct a normal at point P, first find the tangent line at P by determining the slope of the curve at that point. The normal will be perpendicular to this tangent, so use the negative reciprocal of the tangent slope to find the normal line equation.
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