A conic section, also referred to just as a 'Conic' is a curve obtained by intersecting a plane with a cone. Conic sections are the curves obtained by intersecting a plane with a double right circular cone. Imagine a cone being cut by a knife at different places creating different types of curves, which are known as Conic Sections.
The four main Conic sections are: Circle, Ellipse, Parabola, and Hyperbola.
Each type of conic section has unique properties and equations, making them essential for understanding orbital mechanics, designing optical systems, and solving quadratic equations
Formation of Conic Sections
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".
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.
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.
Focus
The focus of a conic section is the point that is used to define various conic sections. The focus of a conic section is different for different conic sections, i.e. a parabola has one focus, while an ellipse and hyperbola have two foci.
Directrix
A line in the 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, a parabola has 1 directrix, ellipse and hyperbola have 2 directrices each.
Eccentricity
The Eccentricity of a conic section is the constant ratio of the distance of the point on the conic section from focus and directrix. We denote eccentricity by the letter "e" and the eccentricity of various conic sections 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
Generated Conic Sections (Sections of Cone)
Depending upon the different angles at which the plane intersects 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 looks like a circle because the top view of an inverted cone is a circle, which 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 make an angle α then various curves formed by the intersection of cone and plane are added below.
(The plane makes the angle β with the cone)
1. Circle
The circle is a conic section in which it is the locus of the point that is always equidistant from the center of one point. 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,
- The general equation of the circle is, (x - h)2 + (y - k)2 = r2.
- Coordinates of Focus: The circle's focus is its center, (h, k)
- Directrix: Not applicable, as circles do not have directrices.
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. We define the parabola as the locus of all the points where the sum of distance from two fixed points (focus) is always in contact. The image for the same is added below,
- Standard Equation:
- Horizontal Major Axis: \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
- Vertical Major Axis: \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1
- Coordinates of Foci:
- Horizontal ellipse: (h \pm c, k)
- Vertical ellipse: (h, k \pm c)
- Where c = \sqrt{a^2 - b^2}
- Equations of Directrices:
- Horizontal ellipse: x = h \pm \frac{a^2}{c}
- Vertical ellipse: y = k \pm \frac{a^2}{c}
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 parabola is the locus of a point that moves in such a way that its distance is always the same distance from a fixed point (called Focus) and a given Line (called Directrix). The image for the same is added below,
- Standard Equation:
- Vertical Parabola: (x - h)^2 = 4p(y - k)
- Horizontal Parabola: (y - k)^2 = 4p(x - h)
- Coordinates of Focus:
- Vertical parabola: (h, k + p)
- Horizontal parabola: (h + p, k)
- Equations of Directrix:
- Vertical parabola: y=k−p
- Horizontal parabola: x=h−p
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. We define the hyperbola as the locus of a point where the ratio of distance from a fixed point (focus) and a fixed line (directrix) is always constant. The image for the same is added below,
- Standard Equation:
- Horizontal Transverse Axis:\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
- Vertical Transverse Axis: \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1
- Coordinates of Foci:
- Horizontal hyperbola: (h \pm c, k)
- Vertical hyperbola: (h, k \pm c)
- Where c = \sqrt{a^2 + b^2}
- Equations of Directrices:
- Horizontal hyperbola: x = h \pm \frac{a^2}{c}
- Vertical hyperbola: y = k \pm \frac{a^2}{c}
Conic Sections Parameters
Various parameters of the conic section that are used to explain and trace various conic sections are,
- Principal Axis: A line passing through the center and the foci of a conic is called the principal axis, it is also called the major axis of the conic.
- Conjugate Axis: Conjugate axis is the axis that is perpendicular to the principal axis and passes through the center of the conic. It is also called the minor axis.
- Center: The 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 cuts the axis.
- Focal Chord: In a conic section focal chord is the chord passing through the 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.
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 center. The vertices are (±a, 0), and the foci are (±c, 0). For the standard form, the conic section always passes through the origin. The standard forms of the various conic sections 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 center is Origin (0, 0) | Equation when center 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) |
Check - Identifying Conic Sections from their Equation
Conic Sections in Real Life
Various instances where we use the conic sections in our real life include,
- Various shapes around us such as cakes, tables, plates, etc. all are circular in nature.
- Orbits of planets around the sun are elliptical in nature.
- Telescopes and Antennas designed to observe the outer spaces have hyperbolic Mirrors and Lenses.
- The path of projectile motion is defined using a parabola, etc.
Articles Related to Applications of Conic Sections-
Articles related to Conic Sections
Solved Examples of Conic Sections
Example 1: Find the equation of a circle that has a center 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 center (-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 center: x2 + 6x + y2 - 4y = 3
Solution:
We are given the equation, now to find out the radius and the center. 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 center (-h,-k) and radius \sqrt{h^2 + k^2}
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 center of the circle lies on the line. Find out the radius and center 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 center 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
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