A bio-medical device, as shown in the graphic below, generates sound waves at one focus of an ellipse - Leaving Cert DCG - Question A-2 - 2011
Question A-2
A bio-medical device, as shown in the graphic below, generates sound waves at one focus of an ellipse. The waves are then reflected to the other focus to shatter a p... show full transcript
Worked Solution & Example Answer:A bio-medical device, as shown in the graphic below, generates sound waves at one focus of an ellipse - Leaving Cert DCG - Question A-2 - 2011
Step 1
Locate 2nd focus (2,1).
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Answer
The second focus of the ellipse is located at the point (2, 1). This is determined based on the property that foci of an ellipse are situated symmetrically with respect to the center.
Step 2
Determine points on curve ... (min. 5, to include end of minor axis 'a').
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To determine points on the curve, we can use the standard equation of the ellipse and compute at least five points. Starting from the semi-major and semi-minor axes, we can find points using parameterization. For example, using angles such as 0, 30, 60 degrees to find points like (a, b), keeping the focus at (2, 1).
Step 3
Draw curve (Any = 1).
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To draw the top half of the curve, we smoothly connect the determined points while ensuring the curve exhibits the elliptical shape, ensuring symmetry about the major and minor axes.
Step 4
Identify point 70mm from F.
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To identify a point that is 70mm from focus F, we measure the distance from the point F along the directrix horizontally. Based on the scale of the graph, we mark a point accordingly.
Step 5
Required construction and draw tangent (3,1).
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To draw the tangent at the identified point (3, 1), we first construct the radius to that point and then apply the derivative of the curve to find the slope. After establishing the slope, we draw the tangent line extending it in both directions.
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