Section of Cone: Introduction, Parameters, Examples, Degenerate, Cross-section

Section of cone: A cone is a three-dimensional object with a circular base, a circular lateral surface, and a top point. The cone is formed by revolving the right triangle along with its height. The terms apex and vertex refer to the same location. A conic section is a section generated by intersecting a plane with a cone.

When a plane intersects a right circular cone, conic sections are created. We get many different cross-sections when we slice an object with a parallel plane. Similarly, depending on how we cut a right circular cone into pieces, we can obtain a variety of cross-sections. A cone has four potential cross-sections: a circle, an ellipse, a parabola, and a hyperbola.

Generation of Cone

A cone is a solid three-dimensional geometric object with a circular base and a pointed apex at the top. A cone is made up of one face and one vertex. Generally, the cone is formed by revolving the right triangle along its height.

Definition of Section of Cone

Let \(l\) be a fixed vertical line, and \(m\) be a line that intersects it at a given point \(V\) and is inclined at an angle, \(\alpha \).

Assume that we rotate \(m\) around \(l\) while keeping \(\alpha \) constant. The resulting surface is a double-napped right-circular hollow cone, referred to as cone hereafter, that extends infinitely in both directions.

The vertex is the point \(V\), and the axis of the cone is the line \(l\). The revolving line \(m\) is known as the cone generator. The nappes are the two segments of the cone that are separated at the vertex. Conic sections are the curves formed when a plane intersects a right circular cone.

What are the Sections of a Cone?

Depending on the angle of the cut, we can get a variety of shapes such as:

• Circle
• Ellipse
• Parabola
• Hyperbola

• A circle is an ellipse in which the cutting plane is parallel to the base of the cone.
• An ellipse is a conic section formed when a plane intersects at an angle, cutting both the slanting sides of the cone.
• When the plane intersects with both nappes of the cone parallel to the axis of the cone, a hyperbola is formed.
• We get a parabola conic section when the plane intersects the base and one of the slanting sides of the cone.

Circle

• The circle has a focal point known as the circle’s centre.
• The radius of the circle is equal to the distance between the loci of the points on the circle and the focus or centre of the circle.
• For a circle, the eccentricity, \(e=0\).
• There is no directrix in a circle.
• The general form of a circle’s equation with centre at \(\left( {h,k} \right)\) and radius \(r:{\left( {x – h} \right)^2} + {\left( {y – k} \right)^2} = {r^2}\)

Parabola

• A parabola is a conic section formed when the intersecting plane is at an angle to the cone’s surface.
• It is a conic section in the shape of a \({\mathbf{U}}\).
• Eccentricity of a parabola, \(e=1\).
• It is an asymmetrical open plane curve generated when a cone meets a plane parallel to one of its sides.

Hyperbola

• When the intersecting plane is parallel to the axis of the cone and intersects both nappes of the double cone, a hyperbola is formed.
• For hyperbola, the eccentricity \(\left( e \right)\) value is \(e > 1\).
• Branches are the two unconnected sections of the hyperbola.
• Their diagonally opposing arms reach the boundary, and they are mirror reflections of each other.

Ellipse

• Ellipse is a conic section generated when a plane intersects a cone at an oblique angle.
• There are two foci, a major axis, and a minor axis in an ellipse.
• For an ellipse, the eccentricity, \(e < 1\).
• Ellipse has two axes of rotation.
• The general form of an elliptical equation with the centre at \(\left( {h,k} \right)\) and the major and minor axis lengths of ‘\(2a\)’ and ‘\(2b\)’, respectively.
• The ellipse’s major axis is parallel to the \(x-\)axis.

PRACTICE EXAM QUESTIONS AT EMBIBE

Parameters of a Conic Section

There are various parameters such as focus, eccentricity, and directrix.

1. Focus of a Section of Cone

The focus of a conic section or foci (plural) is the point around which the conic section is formed.

2. Directrix of a Section of Cone

A directrix is a line that is used to define conic sections. The line drawn perpendicular to the axis of a conic is called the directrix.

3. Eccentricity of a Section of Cone

The eccentricity of a conic section is the constant ratio of the distance of a point from the focus and that from the directrix. It is used to define the shape of a conic section uniquely. The letter \(e\) is used to represent eccentricity. The values of \(e\) for various conic sections are as follows.

• For circle, \(e=0\).
• For ellipse, \(0 \leqslant e < 1\)
• For parabola, \(e=1\)
• For hyperbola, \(e>0\) 

Degenerate Conic

A degenerate conic is obtained when the plane cuts through the vertex of the cone. It is a plane curve of second-degree and is defined by a polynomial equation of the same degree. It can be a single line, two lines that are either parallel or not, a single point, or a null set. They belong to the family of geometric objects that share the property of being conics.

We can define these conic sections’ equations in standard form by using geometric properties.

When the plane cuts at the vertex of the cone, we have the following different cases:

• When \(\alpha  < \beta  \leqslant {90^ \circ }\), then the section is a point.
• When \(\beta  = \alpha \), the plane contains a generator of a cone, and the section is a straight line. It is the degenerated case of a parabola.
• When \(0 \leqslant \beta  < \alpha \), the section is a pair of intersecting straight lines. It is the degenerated case of a hyperbola.

Formulas of Sections of Cone

The standard forms of a parabola, circle, ellipse, and hyperbola are represented by conic section formulas.

The standard form of ellipses and hyperbolas has the \(x-\)axis as the principal axis and the origin \(\left( {0,0} \right)\) as the centre. The vertices are \(\left( { \pm a,0} \right)\) and the foci are \(\left( { \pm c,0} \right)\) It is defined by \({c^2} = {a^2} – {b^2}\) for an ellipse and \({c^2} = {a^2} + {b^2}\) for a hyperbola. Since \(c=0\) for a circle, \({a^2} = {b^2}\). The standard form of the parabola has the focus at the point \(\left( {a,0} \right)\) on the \(x-\)axis, and the directrix is the line with the equation \(x=-a\).

• Ellipse: \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)
• Hyperbola: \(\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1\)
• Circle: \({x^2} + {y^2} = {a^2}\)
• Parabola: \({y^2} = 4ax\) when \(a > 0\)

Uses of Sections of Cone

Conics are significant because they provide simple and useful mathematical relationships. It can be found in a variety of forms in nature. In Physics, it is commonly used:

• To understand light, we must first understand parabolas. If you put a light on a parabolic mirror, all light will concentrate on one point, known as the focus.
• Paths visible around the Sun are ellipses with the Sun at the centre.
• In telescope systems, hyperbolic and parabolic mirrors are used.
• Objects fired near the Earth’s surface typically travel in a parabolic path.

Solved Examples – Section of Cone

Below are a few solved examples that can help in getting a better idea.

Q1. If the focus of an ellipse is at \(\left( {3,0} \right)\) the vertex is at \(\left( {4,0} \right)\) and the centre is at \(\left( {0,0} \right)\) Find the ellipse’s equation.
Solution:
Given:
Focus\( = \left( {3,0} \right)\)
Vertex\( = \left( {4,0} \right)\)
Centre\( = \left( {0,0} \right)\)
So, \(c=3\)
\(a=4\)
Now, \({b^2} = {a^2} – {c^2}\)
\( = {4^2} – {3^3}\)
\( = 16 – 9\)
\(\therefore \,\,{b^2} = 7\)
The equation of an ellipse is given  by
\(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)
\(\frac{{{x^2}}}{{{4^2}}} + \frac{{{y^2}}}{{{7}}} = 1\)
\(\frac{{{x^2}}}{{{16}}} + \frac{{{y^2}}}{{{7}}} = 1\)

Q2. Which section of cone is represented by the equation \({y^2} – 8y = 4x + 4\)?
Solution:
Given equation is \({y^2} – 8y = 4x + 4\)
\({y^2} – 8y + 16 – 16 = 4x + 4\)
\({y^2} – 2\left( 4 \right)\left( y \right) + {4^2} = 4x + 4 + 16\)
\({\left( {y – 4} \right)^2} = 4x + 20\)
\({\left( {y – 4} \right)^2} = 4\left( {x + 5} \right)\)
\({y^2} = 4ax\)
So, the given equation represents a parabola.

Q3. Find the eccentricity of the section of a cone, which is represented by equation \(16{x^2} + 25{y^2} = 400.\)
Solution:
Given: \(16{x^2} + 25{y^2} = 400\)
\(\frac{{16{x^2}}}{{400}} + \frac{{25{y^2}}}{{400}} = \frac{{400}}{{400}}\)
\(\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1\)
\(\frac{{{x^2}}}{{{5^2}}} + \frac{{{y^2}}}{{{4^2}}} = 1\)
The equation is in the form of an ellipse:  \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)
Here, \(a=5\), \(b=4\)
Eccentricity is given by
\(e = \sqrt {1 – \frac{{{b^2}}}{{{a^2}}}} \)
\( = \sqrt {1 – \frac{{16}}{{25}}} \)
\( = \sqrt {\frac{{25 – 16}}{{25}}} \)
\( = \sqrt {\frac{9}{{25}}} \)
\(\therefore \,e = \frac{3}{5}\)
Hence, the eccentricity of the given section of the cone is \(\frac{3}{5}\).

Q4. The equation \({x^2} + {y^2} + 6x + 8y + 21 = 0\) represents which section of the cone?
Solution:
Given: \({x^2} + {y^2} + 6x + 8y + 21 = 0\)
\({x^2} + {y^2} + 6x + 8y + 21 + 4 – 4 = 0\)
\({x^2} + {y^2} + 6x + 8y + 9 + 16 = 4\)
\({x^2} + 6x + 9 + {y^2} + 8y + 16 = 4\)
\({\left( {x + 3} \right)^2} + {\left( {y + 4} \right)^2} = 4\)
This equation is the equation of a circle whose centre is \(\left( {-3,-4} \right)\)

Q5. What is the equation for a hyperbola with a centre at \(\left( {2,3} \right)\), a vertex at \(\left( {0,3} \right)\), and a focus at \(\left( {5,3} \right)\).
Solution:
Given:
Focus\( = \left( {5,3} \right)\)
Vertex\( = \left( {0,3} \right)\)
Centre\( = \left( {2,3} \right)\)
All lie on the line \(y=3\)
Comparing all, \(a=2\), \(c=3\)
Thus \({b^2} = {c^2} – {a^2}\)
\( = {3^2} – {2^2}\)
\( = 9 – 4\)
\(\therefore \,\,{b^2} = 5\)
Equation of a hyperbola is given by
\(\frac{{{{\left( {x – h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y – k} \right)}^2}}}{{{b^2}}} = 1\)
\(\frac{{{{\left( {x – 2} \right)}^2}}}{4} + \frac{{{{\left( {y – 3} \right)}^2}}}{5} = 1\)

Summary

Cone is a three-dimensional object with a circular base, and laterals meet at vertex point. It is formed by revolving a right-angled triangle. We can obtain a variety of cross-sections depending on how we cut a right circular cone into pieces. A parabola, hyperbola, and ellipse are the three primary sections of a cone or conic section.

The point around which the conic section is formed is the focus, plural foci. The line drawn perpendicular to the axis of a conic is called the directrix. The ratio of the distance of a point from the focus and that from the directrix is called eccentricity. A degenerate conic is obtained when the plane cuts through the vertex of a cone.

FAQs on Section of Cone

Students might be having many questions with respect to the Section of Cone. Here are a few commonly asked questions and answers.

Q.1. What are the applications of the section of a cone?
Ans: Conic sections are helpful in Arithmetic, Physics, and Astronomy and a wide range of engineering applications. Conic sections have significant applications in fields like Aerodynamics, where a smooth surface is required to ensure laminar flow and prevent turbulence.

Q.2. How many sections does a cone have?
Ans: A section of a cone is a curve formed by intersecting two napped right-circular cones with a plane. There are four types of sections of cone: circle, ellipse, parabola, and hyperbola.

Q.3. What is a degenerate cone?
Ans: A degenerate conic is obtained when the plane cuts through the vertex of a cone.

Q.4. What are the four sections of a cone?
Ans: Conic sections are the shapes formed when a plane intersects a double cone, as shown below. In other words, conic sections are the cross-sections of a double cone. The four fundamental conic sections are the circle, the parabola, the ellipse, and the hyperbola.

Q.5. What is the eccentricity of the section of a cone?
Ans: A conic section’s eccentricity is the constant ratio of the distance of a point on the conic section from the directrix and that from the focus. It is a non-negative real number ranging from \(0\) to \(1\).

• For circle, \(e=0\).
• For ellipse, \(0 \leqslant e < 1\)
• For parabola, \(e=1\)
• For hyperbola, \(e>0\) 

ATTEMPT MOCK TESTS ON EMBIBE

We hope this information about the Section of Cone has been helpful. If you have any doubts, comment in the section below, and we will get back to you soon.