Sections of a Cone: Definition, Properties & Examples

Sections of a Cone: A cone is a three-dimensional object with a flat base, a circular lateral surface, and a point at the top. Apex or vertex are two terms for the same place. Conical refers to objects that have a cone shape. When we slice an object with a plane, we get a lot of parallel cross-sections.

Similarly, depending on how we cut a right circular cone into pieces, we can get different cross-sections out of it. Circle, Ellipse, Parabola, and Hyperbola are the four potential cross-sections of a cone. Let’s learn about sections of a cone in detail.

What are Sections of a Cone?

We receive a section when we intersect a plane with the right circular cone, and that section is known as a conic section. Alternatively, we can state that if we slice a \(3\text {D}\) cone with a plane from various angles, we will receive different sections, referred to as conic sections or sections of a cone.

Thus, a conic section is a curve created with the intersection of two napped right circular cones and a plane. Conic sections are divided into four types: circle, ellipse, parabola, and hyperbola.

The type of conic section we get is determined by the location of the plane in relation to the cone and the angle formed between the vertical axis and the intersecting plane.

Let the semi-vertical angle of a cone is \(\theta\), and the angle between intersecting plane & the vertical axis be \(\emptyset\). If the plane slices the cone’s one nappe in such a way that

1. \(\theta<\emptyset<90^{\circ}:\) an ellipse is formed when a plane meets a double circular cone in such a way that the angle between the axis and the plane is greater than the vertex angle.
2. \(\emptyset=90^{\circ}:\) the ellipse becomes a circle when the plane is perpendicular to the axis or when the angle of intersection of the plane and the vertical axis is equal to \(90^{\circ}\). Or, to put it another way, a circle is a special form of an ellipse, and its section is a circle.
3. \(\theta=\emptyset:\) at the intersecting surface of the cone, we get an open curve called a parabola when the angle between the vertical axis and the plane equals the vertex angle.
4. \(\emptyset<\theta:\) a hyperbola is formed when a plane cuts the two nappes of a cone in such a way that the plane intersects both nappes of the cone, forming two disjoint open curves when the angle between the plane and the vertical axis is smaller than the vertical angle.

We can identify the sections of a cone when its general equation is given:

The general equation of a conic section looks like \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0\), the above expression will represent different sections of a conic if

1. \(\Delta \neq\ 0\) and \(a=b \neq 0 \,\&\, h=0 \Rightarrow\) The above general equation will be a circle
2. \(\Delta \,\neq\ 0\) and \(h^{2}=a b \Rightarrow\) The above general equation will be a parabola
3. \(\Delta \neq 0\) and \(h^{2}<a b \Rightarrow\) The above general equation will be an ellipse
4. \(\Delta \neq 0\) and \(h^{2}>a b \Rightarrow\) The above general equation will be a hyperbola

Where \(\Delta = \left| {\begin{array}{*{20}{c}} a&h&g\\ h&b&f\\ g&f&c \end{array}} \right|\)

The standard equation of different conic sections are as follows:

(i) Standard parabola: \(y^{2}=4 a x\)

(ii) Standard circle: \((x-a)^{2}+(y-b)^{2}=R^{2}\); where centre \((a, b)\) and radius \(=R\)

(iii) Standard ellipse: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,(a>b) ; b^{2}=a^{2}\left(1-e^{2}\right)\); where \(e=\) eccentricity

(iv) Standard hyperbola : \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 ; b^{2}=a^{2}\left(e^{2}-1\right)\); where \(e=\) eccentricity

Uses of Sections of a Cone

Conics are significant because they provide straightforward and practical mathematical relationships. It can be found in nature in a variety of forms. It is also significant in our daily lives. It is commonly used in physics as

1. To learn about light, we must first learn about parabolas. If you shine a light on a parabolic mirror, all of the light will focus on one point, referred to as focus.
2. The paths you see around the sun are ellipses with the sun in the centre.
3. Hyperbolic and parabolic mirrors are used in telescope systems.
4. Objects fired near the earth’s surface usually travel in a parabolic direction.
5. The parabola is often used to construct automotive headlights.

All conics play a significant part in natural physical processes. Conic sections play a crucial role in everything from the universe to the atom.

Parameters of Conic Sections

A conic section is defined as the locus of a moving point in the plane such that the ratio of its distance from a fixed point to its distance from a fixed line equals a constant. The fixed point is known as a Focus, and the fixed line is known as directrix, and the constant is known as eccentricity \((e)\).

If \(e=0\), the conic section will be a circle.
If \(0<e<1\), the conic section will be an ellipse
If \(e=1\), the conic section will be a parabola
If \(e>1\), the conic section will be a hyperbola

In general, eccentricity refers to the degree to which a curve has deviated from the circularity of a given shape.

There are a few other parameters described under conic sections besides focus, eccentricity, and directrix. They are 

1. The line connecting the two focal points or foci of an ellipse or hyperbola is known as the Principal axis. The curve’s centre is halfway.
2. A chord parallel to the directrix that goes through a focus is known as the Latus Rectum.
3. The distance between the focus and the corresponding directrix is known as the Focal parameter.
4. Linear eccentricity is the distance between a section’s focus and its centre.
5. The point of intersections of any conic with the principal axis are called Vertices.

Solved Examples – Sections of a Cone

Q.1. Which conic section is represented by \(9 x^{2}-16 y^{2}=144\)?
Ans: Given, \(9 x^{2}-16 y^{2}=144\)
\(\Rightarrow \frac{x^{2}}{16}-\frac{y^{2}}{9}=1\)
On comparing with the standard equation of hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), we get \(a=4\) and \(b=3\)
Hence, The above equation represents the standard equation of hyperbola

Q.2. What will be the eccentricity of ellipse \(16 x^{2}+25 y^{2}=400\)?
Ans: Given, \(16 x^{2}+25 y^{2}=400\)
\(\Rightarrow \frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)
Here, \(a=5\) and \(b=4\)
So, \(e=\sqrt{1-\frac{b^{2}}{a^{2}}}=\sqrt{1-\frac{16}{25}}=\frac{3}{5}\)

Q.3. Which conic section is represented by the equation \(y^{2}-8 y=4 x+4\)?
Ans: Given, \(y^{2}-8 y=4 x+4\)
\(\Rightarrow(y-4)^{2}=4 x+20\)
\(\Rightarrow(y-4)^{2}=4(x+5)\)
The above equation can be seen as the standard equation of a parabola

Q.4. Determine the type of conic for the equation \(9 x^{2}+4 y^{2}+18 x-16 y+24=0\)
Ans: So, here we have:
\(9 x^{2}+4 y^{2}+18 x-16 y+24=0 \Rightarrow\left(x^{2}+2 x\right)+\frac{4}{9}\left(y^{2}-4 y\right)=-\frac{8}{3}\)
\(\Rightarrow \frac{(x-(-1))^{2}}{\left(\frac{1}{3}\right)^{2}}+\frac{(y-2)^{2}}{\left(\frac{1}{2}\right)^{2}}=1\)
Hence, On comparing it with the standard equation of conics, We got to know it is an ellipse.

Q.5. Find the centre of the conic for the equation \(x^{2}+y^{2}+6 x+8 y+21=0\)
Ans: \(x^{2}+y^{2}+6 x+8 y+21=0\)
\(\Rightarrow(x+3)^{2}+(y+4)^{2}=4\)
This is the equation of the circle, and so its centre of the circle will be \((-3,-4)\)

Summary

In this article, we have learnt about the sections of a cone & its parameter and solved examples. A conic section is a curve created by the intersection of two napped right circular cones and a plane. Conic sections are divided into four types: circle, ellipse, parabola, and hyperbola.

We can easily identify the conic sections when their general equations are given. We have also learnt about the parameters which are used in conic sections. With the knowledge of parameters, we can clearly understand the conic in a better way.

Learn All the Concepts on Cones

Frequently Asked Questions (FAQs) – Sections of a Cone

Q.1. How do you identify a conic section?
Ans: The general equation for any conic section is \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0\).
1. If \(\Delta \neq 0\) and \(a=b \neq 0\, \& \,h=0 \Rightarrow\) The above general equation will be a circle
2. If \(\Delta \neq 0\) and \(h^{2}=a b \Rightarrow\) The above general equation will be a parabola
3. If \(\Delta \neq 0\) and \(h^{2}<a b \Rightarrow\) The above general equation will be an ellipse
4. If \(\Delta \neq 0\) and \(h^{2}>a b \Rightarrow\) The above general equation will be a hyperbola
Where \(\Delta = \left| {\begin{array}{*{20}{c}} a&h&g\\ h&b&f\\ g&f&c \end{array}} \right|\)

Q.2. What are the 4 conic sections?
Ans: Conic sections are the shapes that can be made when a plane intersects a double cone, such as the one seen below. Conic sections, in other terms, are the cross-sections of a double cone. The circle, the parabola, the ellipse, and the hyperbola are the four fundamental conic sections.

Q.3. What are the examples of sections of a cone?
Ans: The real-life examples where sections of a cone play an important role are as follows:
1. The planets’ orbits around the sun are ellipses, with the sun at the centre.
2. Light beams are converged at the parabola’s focus using parabolic reflectors.
3. Hyperbolas are utilised in the LORAN navigation system (long-range navigation)
4. Parabolic mirrors are used in solar ovens to focus light beams for heating.
5. In telescope systems, hyperbolic and parabolic mirrors and lenses are used.

Q.4. How do you cut a conic section?
Ans: Conic sections are obtained by cutting a right circular cone by a plane. The different orientation of plane gives different types of conics. They are the circle, the ellipse, the parabola and the hyperbola

Q.5. What are the four types of conic sections?
Ans: A conic section is a curve created by the intersection of two napped right circular cones and a plane. Conic sections are divided into four types: circle, ellipse, parabola, and hyperbola.

Now you are provided with all the necessary information on sections of a cone and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.