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Conic Sections: Circle, Parabola, Ellipse, Hyperbola - Equation & Example

Conic Sections in Geometry: Conic Sections also called the section of the cone are the curves that are formed by the intersection of a plane and a cone. Thus, the name Conic Sections because they are the section of a cone. By the intersection of a cone with a plane, we formed four different types of conic sections namely,

  • Circle
  • Parabola
  • Hyperbola
  • Ellipse

In this article, we have covered What are Conic Section, Conic Section Formulas, Equations of Conic Sections, and others in detail.

conic sections

Table of Content

What are Conic Sections?

A Conic section, also referred to just as a ‘Conic’ is a plane intersecting a cone. Imagine a cone being cut by a knife at different places creating different types of curves, which are known as Conic Sections. The three main Conic sections Parabola, Hyperbola, and Ellipse (a Circle can be referred to as a type of ellipse).

Conic Sections Definition

Conic sections are the curves obtained by intersecting a plane with a double right circular cone.

Let’s say we take a fixed vertical line. We’ll call it “l”. Now make another line at a constant angle α from this line as shown in the image added below, the other line is “m”.

cone-generating-lines

Now if we start rotating the line m around l by keeping the angle the same. We will get a cone that extends to infinite in both directions. 

Double Napped Cone

The rotating line(m) is called the generator of the cone. The vertical line(l) is the axis of the cone. V is the vertex, it separates the cone into two parts called nappes. 

Now when we take the intersection of the generated cone with a plane, the section obtained is called a conic section. This intersection generates different types of curves depending upon the angle of the plane that is intersecting with the cone. These different types of curves and their image added in the below:

Generated Conic Sections (Sections of Cone)

Depending upon the different angles at which the plane is intersecting the Cone, different types of curves are found. Imagine that an ice cream cone is in the hands, looking at the cone from the top it look like a circle because the top view of an inverted cone is a circle, that gives a conclusion that cutting a cone with a plane exactly at 90° will provide a circle, Similarly, different angles will lead to different types of curves.

Let’s see that plane makes an angle β with the vertical axis. Depending on the value of the angle there can be several curves of intersections. Suppose the vertical line and the generator line of the conic section makes an angle α then various curves formed by intersection of cone and plane are added below, now let’s learn about them in detail.

(The plane makes the angle β with the cone)

1. Circle

If the plane cuts the conic section at right angles, i.e. β = 90° then we get a circle. The image for the same is added below,

Circles

2. Ellipse

If the plane cuts the conic section at an angle less than 90°, i.e. α < β < 90° then we get an ellipse. The image for the same is added below,

Ellipse

3. Parabola

If the plane cuts the conic section at an angle where α is equal to β i.e. α = β then we get a parabola. The image for the same is added below,

Parabola

4. Hyperbola

If the plane cuts the conic section at an angle where β is less than α i.e. β ϵ [0, a] then we get a hyperbola. The image for the same is added below,

Hyperbola

Focus, Eccentricity and Directrix of Conic Section

We define conic section as the locus of a point say (P) moving in the plane about a fixed point F (i.e. the Focus) and with respect to a fixed line known as Directrix (such that the focus point is never on d) and all of these are arranged such that the ratio of the distance of point P from focus F to the distance from d is always contact and the ratio is called the eccentricity(e).

Now lets learn about them in detail.

Focus

Focus of a conic section is the point that is used to define various conic section. The focus of a conic section is different for different conic sections, i.e. a parabola has one focus, while ellipse and hyperbola has two foci.

Directrix

A line in conic section that is perpendicular to the axis of the referred conic is called the directrix of the conic. The directrix of the conic is parallel to the conjugate axis and the latus rectum of the conic. The directrix varies for various conic sections. A circle has no directrix, parabola has 1 directrix, ellipse and hyperbola have 2 directrices each.

Eccentricity

Eccentricity of a conic section is the constant ratio of the distance of the point on conic section from focus and directrix. We denote eccentricity by letter “e” and the eccentricity of various conic section are,

  • For e = 0 the conic section is Circle
  • For 0 ≤ e < 1 the conic section is Ellipse
  • For e = 1 the conic section is Parabola
  • For e > 1 the conic section is Hyperbola

Conic Sections Parameters

Variou parameters of the conic section that are used to explain and trace various conic section are,

  • Principal Axis: A line passing through the center and the foci of a conic is called the principal axis, it is also called major axis of the conic.
  • Conjugate Axis: Conjugate axis is the axis that is perpendicular to the principal axis and passing through the center of the conic. It is also called the minor axis.
  • Center: Center of the conic is defined as the point of intersection of the principal axis and the conjugate axis.
  • Vertex: Vertex of the conic is defined as the point of the principal axis where the conic cut the axis.
  • Focal Chord: In a conic section focal chord is the chord passing through focus of the conic section.
  • Latus Rectum: A focal chord perpendicular to the axis of the conic section is called the latus rectum of the conic.

Conic Section Formulas

Various Conic Section Formulas that are associated to the conic section are added in the table below,

Conic Section

Formula

Parameter

Circle

(x − a)2 + (y − b)2 = r2

Center is (a,b)

Radius is r

Ellipse

(x − a)2/h2 + (y − b)2/k2 = 1

Center is (a, b)

c2 = h2 − k2, h > k > 0

Parabola

(y − b)2 = 4p(x − a), p≠0

Vertex is (a, b)

Focus is (a+p, b)

Hyperbola

(x − a)2/h2 − (y − b)2/k2 = 1

Center is (a, b)

c2 = h2 − k2, h > k > 0

Types of Conic Sections

Intersectiong a plane with the cone we get different types of the conic sections. There are four general types of conic section that are,

Circle

The circle is a conic section in which it is the locus of the point that is always equidistant from the centre of one point. The general equation of the circle is, (x – h)2 + (y-k)2 = r2

Parabola

We define the parabola as the locus of a point that moves in such a way that its distance is always same distance from a fixed point (called Focus) and a given Line (called Directrix). The general equation of the parabola is, y = a(x-h)2 + k

Hyperbola

We define the hyperbola as the locus of a point that the ratio of distance of from a fixed points (focus) and a fixed line (directreix) is always constant. The general equation of the hyperbola is, [(x2/a2) – (y2/b2)] = 1

Ellipse

We define the parabola as the locus of all the points that the sum of distance from two fixed points (focus) is always contact. The general equation of the ellipse is, [(x2/a2) + (y2/b2)] = 1

Standard Form of Conic Sections

The standard form of the conic section is added below, For ellipses and hyperbolas, the standard form has the x-axis as the principal axis and the origin (0,0) as the centre. The vertices are (±a, 0) and the foci (±c, 0). For the standard form the conic section always passes through the origin. The standard form of the various conic section are,

  • Circle: x2 + y2 = a2
  • Ellipse: x2/a2 + y2/b2 = 1
  • Hyperbola: x2/a2 – y2/b2 = 1
  • Parabola: y2 = 4ax when a > 0

Conic Sections Equations

The standard equations of the conic section are added in the table below,

Conic Section

Equation when Centre is Origin (0, 0)

Equation when Centre is (h, k)

Circle

x2 + y2 = r2; r is radius

(x – h)2 + (y – k)2 = r2; r is radius

Ellipse

(x2/a2) + (y2/b2) = 1

(x – h)2/a2 + (y – k)2/b2 = 1

Hyperbola

(x2/a2) – (y2/b2) = 1

(x – h)2/a2 – (y – k)2/b2 = 1

Parabola

y2 = 4ax

(y – k)2 = 4a(x – h)

Conic Sections in Real Life

Various instances where we use the conic sections in our real life includes,

  • Various shapes around us such as cake, table, and plates, etc all are circular in nature.
  • Orbits of planets around the sun are elliptical in nature.
  • Telescopes and Antennas designed to observe the outerpaces have hyperbolic Mirrors and Lenes.
  • Path of projectile motion is defined using a parabola, etc.

Articles related to Conic Sections

Conic Sections Solved Examples

Example 1: Find the equation of a circle that has a centre of (0,0) and a radius is 5.

Solution: 

We have studied the formula for the equation of the circle. 

(x-h)2 + (y – k)2 = r2

We just need to plug in the values in the formula. 

Here, h = 0, k = 0 and r = 5 

(x – 0)2 + (y – 0)2 = 52

⇒x2 + y2 = 52

⇒x2 + y2 = 25

Example 2: Find the equation of the circle with centre (-4, 5) and radius 4. 

Solution: 

The formula for the equation of the circle. 

(x-h)2 + (y – k)2 = r2

We just need to plug in the values in the formula. 

Here, h = -4, k = 5 and r = 4 

(x – (-4))2 + (y – 5)2 = 52

⇒(x+ 4)2 + (y – 5)2 = 25

⇒x2 + 16 + 8x + y2 + 25 – 10y = 25

⇒x2 + 8x + y2 -10y + 16= 0

Example 3: The equation given below is an equation of the circle, find out the radius and the centre. 

x2 + 6x + y2 – 4y = 3

Solution: 

We are given the equation, now to find out the radius and the centre. We need to rearrange the equation such that this equation can come in the form given below. 

(x-h)2 + (y – k)2 = r2

x2 + 6x + y2 – 4y = 3

⇒ x2 + (2)(3)x + y2 – 2(2)y = 3 

We can see that these equations can be separated into two perfect squares. 

⇒ x2 + (2)(3)x + 9 – 9 + y2 – 2(2)y + 4 – 4 = 3 

⇒ (x + 3)2 – 9 + (y – 2)2 – 4 = 3 

⇒ (x + 3)2 + (y – 2)2 = 3  + 4 + 9 

⇒ (x + 3)2 + (y – 2)2 = 16

⇒ (x + 3)2 + (y – 2)2 = 42

Now comparing this equation with the standard equation of the circle, we notice, 

h = -3, k = 2 and radius = 4. 

Example 4: Find the equation of the circle, with centre (-h,-k) and radius [Tex]\sqrt{h^2 + k^2} [/Tex]

Solution: 

The standard equation of the circle is given by, 

(x-h)2 + (y – k)2 = r2

Here, we have h = -h and k = -k and radius = √{h2 + k2}

Putting these values into the equation

(x + h)2 + (y + k)2 = (√{h2 + k2})2

x2 + h2 + 2hx + y2 + k2 + 2ky = h2 + k2

x2 + y2 + 2hx + 2ky = 0

Example 5: Let’s say we are given a line x + y = 2 and a circle that passes through the points (2,-2) and (3,4). It is also given that the centee of the circle lies on the line. Find out the radius and centre of the circle. 

Solution: 

Let’s say the equation of the circle is, 

(x-h)2 + (y – k)2 = r2

Now we know that the center of the circle lies on the line x + y = 2. Since the centre of the circle is (h, k), it should satisfy this line. 

h + k = 2

Putting the value of h from this equation into the equation of the circle. 

(x-(2 – k))2 + (y – k)2 = r2

Now we also know that the circle satisfies the points (2, -2) and (3,4). Putting (2,-2) in the above equation.

(2-(2 – k))2 + (-2 – k)2 = r2

⇒ k2 + (k + 2)2 = r2

⇒ k2 + k2  + 4 + 4k = r2

⇒ 2k2 + 4 + 4k = r2 …..(1)

Putting the equation (3,4) is, 

(x-(2 – k))2 + (y – k)2 = r2

⇒ (3-(2 – k))2 + (4 – k)2 = r2

⇒(1 – k)2 + (4 – k)2 = r2

⇒ k2 -2k + 1 + 16 -8k + k2 =r2

⇒ 2k2 -10k + 17 =r2 ……(2)

Solving these equations we get, 

h = 0.7, k = 1.7 and r2 = 12.58

Important Maths Related Links:

Conic Sections Class 11 Resources

In Class 11, the study of conic sections is an important part of the mathematics curriculum, particularly in the field of coordinate geometry. Conic sections—circle, ellipse, parabola, and hyperbola—are significant because they provide a foundation for understanding various physical phenomena and have numerous practical applications in real life, such as in the paths of planets and satellites, optics, and engineering designs.

Also Check:

Conic Sections Class 11 Notes
Conic Sections Class 11 NCERT Solutions

Conic Sections Worksheet

Q1: For a hyperbola with vertices (±2, 0) and foci at (±3, 0). Find the equation of the hyperbola.

Q2: Find the equation of the parabola with vertex at origin and focus at (2, 0).

Q3. Find the equation of circle with radius 5 units and center at (1, 1).

Q4. Find the equation of circle with end points of diameter to be (2, 3) and (-4, 6).

FAQs on Conic Sections

What is Conic Section in Geometry?

If we intersect a cone and a plane then the curves so formed are called the conic section in geometry. There are four basic conic section in geometry that include, Ellipse, Circle, Parabola, and Hyperbola.

What is Equation of Parabola?

The standard equation of the Parabola is,

y2 = 4ax

What is Equation of Circle?

The standard equation of the Circle is,

x2 + y2 = r2

What is Equation of Hyperbola?

The standard equation of the Hyperbola is,

(x2/a2) – (y2/b2) = 1

What is Equation of Ellipse?

The standard equation of the Ellipse is,

(x2/a2) + (y2/b2) = 1

What are the 4 Conic Section?

The four conic sections formed by the plane and the cone are,

  • Circle
  • Ellipse
  • Parabola
  • Hyperbola

 


Reffered: https://www.geeksforgeeks.org

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