The image below shows the Spinnaker Tower in Portsmouth - Leaving Cert DCG - Question A-2 - 2021
Question A-2
The image below shows the Spinnaker Tower in Portsmouth. The design includes two hyperbolas. The drawing on the right shows the axis AA₁, directrix DD₁, focus F and ... show full transcript
Worked Solution & Example Answer:The image below shows the Spinnaker Tower in Portsmouth - Leaving Cert DCG - Question A-2 - 2021
Step 1
Locate the vertex
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Answer
The vertex of the hyperbola can be found at the point V, which is equidistant from the focus F and the directrix D₁. Using the given diagram, the coordinates for the vertex V can be determined accurately.
Step 2
Location of 2 points outside latus rectum
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Answer
Two points outside the latus rectum can be determined based on the relationship between the hyperbola's directrices and foci. Using the focal distance and the properties of the hyperbola, we can find two points, for instance, at coordinates (x₁, y₁) and (x₂, y₂).
Step 3
Location of 3 points inside latus rectum
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Three points inside the latus rectum can be calculated by determining equidistant points along the axis of symmetry that lies between the directrix and the focus. Consider points such as (x₃, y₃), (x₄, y₄), and (x₅, y₅).
Step 4
Draw curve (any = 1)
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To sketch the hyperbola, begin from the vertex and plot the points calculated previously. Connect these points smoothly to illustrate the top portion of the hyperbola, ensuring the appropriate asymptotic behavior is represented.
Step 5
Draw tangent to the curve at point P
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To draw the tangent at point P on the hyperbola, use the derivative of the hyperbola's equation. At point P, find the slope of the tangent line using the formula, and then draw a line through point P with that slope.
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