Here we will understand the concept of different conic sections that can be obtained using a double napped cone. A double napped cone can be formed by taking two identical right circular cones inverting one of them, and then joining the two cones at their vertices.
It can also be formed by taking a straight line at a fixed angle with another straight line called the axis, and rotating it through an angle of $360^\circ$ about that axis.
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The double-napped cone shown in the above widget is drawn by taking a line at $45^\circ$ with the $y$-axis and then rotating it by an angle of $360^\circ$ about the $y$-axis.
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The plane shown in the widget can be rotated to cut the double napped cone at various angles. You can try rotating the plane, using the circular slider and observe the shape of the cut section of the cone.
Note, that all measurements related to eccentricity of the conic sections obtained are in terms of the angle that the edge of the cone makes with the $y$-axis ($\alpha$) and the angle the cutting plane makes with the $y$-axis ($\beta$). For better visibility the widget shows the complementary angles $\alpha '$ and $\beta '$ with the $x$-axis, therefore, $\alpha ' = 90^\circ - \alpha$ and $\beta ' = 90^\circ - \beta$
You can try rotating the given plane at various angles to see how the section of the cone cut by the plane looks like.
The eccentricity, $(e)$, of the generated conic section is defined as
You should be able to make the following observations from your experiments with the given widget.
- When the cutting plane is perpendicular to the $y$-axis, that is, $|\beta | = 90^\circ$ you get a circle, and the eccentricity is $0$
- When the cutting plane is parallel to any of the edges of the cone, that is, $|\beta| - |\alpha| = 0$, you get a parabola. Since the cutting plane is parallel to the edge the cut section is closed on one side (the visible end), while on the other side (the infinity side) it is open.
- When the cutting plane forms an angle with the $y$-axis whose magnitude is greater than the angle formed by the side of the cone, that is, $|\beta| \gt |\alpha|$, the plane intersects either the upper cone or the lower cone, but not both, and the curve that we get is an ellipse. Here, $0 \lt e \lt 1$
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- When the cutting plane forms an angle with magnitude less than that of the angle of the edge with the $y$-axis, that is $|\beta| \lt |\alpha|$, the plane intersects both the upper and the lower cones, and the resultant section is a hyperbola. In this case $e \gt 1$
In all of the above cases, we have made one assumption, that is the plane does not pass through the common vertex of the cones. So, what happens when the plane passes through the vertex?
Let us try to answer the following question:
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When the plane passes through the vertex, the section that we obtain is called a degenerate conic section.
A degenerate hyperbola is a pair of intersecting straight lines.
A degenerate ellipse is a point.
A degenerate parabola is a single straight line.
II. Bounded And Unbounded Conic Sections
As we have seen from the previous section, out of all the conic-sections, the ellipse and the circle (which is a special case of ellipse) are bounded on all sides by the line forming the section.
Whereas parabola is a section which is bounded only on one side the other side is open.
The hyperbola has two sections, each of which is bounded on one side and unbounded on the other.
Hence, the parabola and the hyperbola are unbounded conic sections, and the ellipse is a bounded conic section.
III. Conic Sections As Locii
One of the important properties of the conic section is that each can be obtained by plotting the locus of a point moving on a plane such that its distance from a fixed point and from a given straight line not passing through that point are in a fixed ratio.
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This line is is called the directrix and the fixed point is called the focus of the section. As it turns out, this fixed ratio is actually the eccentricity $(e)$ of the conic section.
Hyperbola and ellipse, each have two directrices and two focii. The parabola has a single focus and a directrix. For the circle, since,
$\dfrac{\text{distance from focus}}{\text{distance from directrix}} = e = 0$,
there is no line at a finite distance from any point, which satisfies this condition.
This condition is approached only when the directrix is at infinite distance from the focus, or alternately, the circle itself is a point, and the numerator of the above relation is always zero.
We will look into the properties of each of the conic sections in details in subsequent chapters of this topic.