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The 3D graphic below shows a beam of light shining across a table top and generating a hyperbolic curve - Leaving Cert DCG - Question A-1 - 2009
Question A-1
The 3D graphic below shows a beam of light shining across a table top and generating a hyperbolic curve. The drawing on the right shows the axis, directrix and focu... show full transcript
Worked Solution & Example Answer:The 3D graphic below shows a beam of light shining across a table top and generating a hyperbolic curve - Leaving Cert DCG - Question A-1 - 2009
Step 1
Locate the vertex (1.1)
Answer
To locate the vertex of the hyperbola, we start with its definition. For a hyperbola with a focus located at F and a directrix, the vertex is the point where the curve is closest to the focus. In this case, given the vertical orientation of the hyperbola, the vertex can be plotted as follows:
- Set the value of rac{a}{c} where the eccentricity , such that rac{c}{a} = 1.2.
- Given that denotes the distance from the center to the focus (F), we can calculate the vertex accordingly.
Step 2
Draw curvature (Any = 1)
Answer
To sketch the hyperbola, we can use the standard form equation of the hyperbola centered at the origin. The equation takes the form:
rac{y^2}{a^2} - \frac{x^2}{b^2} = 1
Where the values of 'a' and 'b' can be determined from the calculated distances obtained in the previous steps. The asymptotes can also be drawn through the diagonals of the rectangular box formed by points ( rac{a}{c}, rac{b}{c}) and (- rac{a}{c}, - rac{b}{c}).
Step 3
Draw latus rectum (1.4)
Answer
The latus rectum is the line segment that passes through the focus and is perpendicular to the transverse axis. It has length given by
where 'b' corresponds to the semi-minor axis. Using the previously calculated values of 'a' and 'b', outline this segment in the graph.
Step 4
Determine centre of curvature (1.1)
Answer
The centre of curvature is found at a point where the radius of curvature intersects the normal at the given point on the hyperbola located vertically above the focus. To find this point analytically, apply the formula for the curvature of hyperbolas which modifies to being proportional to the derivatives of the curve at the given point.