Draw an Ellipse by the Auxiliary Circle Method

Ellipse

Ellipse is an integral office of the conic section and is like in properties to a circle. Unlike the circle, an ellipse is oval in shape. An ellipse has an eccentricity less than i, and it represents the locus of points, the sum of whose distances from the 2 foci of the ellipse is a abiding value. A simple example of the ellipse in our daily life is the shape of an egg in a two-dimensional form and the running tracking in a sports stadium.

Hither we shall aim at knowing the definition of an ellipse, the derivation of the equation of an ellipse, and the unlike standard forms of equations of the ellipse.

1.	What is an Ellipse?
2.	Parts of Ellipse
three.	Standard Equations of an Ellipse
4.	Derivation of Ellipse Equation
5.	Ellipse Formulas
6.	Backdrop of an Ellipse
7.	How to Draw an Ellipse?
8.	Graph of Ellipse
nine.	FAQs on Ellipse

What is an Ellipse?

An ellipse in math is the locus of points in a aeroplane in such a style that their distance from a fixed point has a constant ratio of 'e' to its altitude from a fixed line (less than i). The ellipse is a part of the conic section, which is the intersection of a cone with a plane that does not intersect the cone's base. The fixed point is chosen the focus and is denoted by S, the constant ratio 'eastward' as the eccentricity, and the stock-still line is called as directrix (d) of the ellipse.

Ellipse Definition

An ellipse is the locus of points in a plane, the sum of whose distances from two stock-still points is a constant value. The ii fixed points are called the foci of the ellipse.

Ellipse Equation

The full general equation of an ellipse is used to algebraically represent an ellipse in the coordinate aeroplane. The equation of an ellipse can be given equally,

\(\dfrac{x^2}{a^two} + \dfrac{y^two}{b^2} = 1\)

Equation of a Ellipse

Parts of an Ellipse

Permit us get through a few of import terms relating to dissimilar parts of an ellipse.

Focus: The ellipse has 2 foci and their coordinates are F(c, o), and F'(-c, 0). The distance between the foci is thus equal to 2c.
Center: The midpoint of the line joining the two foci is chosen the center of the ellipse.
Major Axis: The length of the major axis of the ellipse is 2a units, and the cease vertices of this major centrality is (a, 0), (-a, 0) respectively.
Small-scale Centrality: The length of the minor axis of the ellipse is 2b units and the end vertices of the pocket-size axis is (0, b), and (0, -b) respectively.
Latus Rectum: The latus rectum is a line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The length of the latus rectum of the ellipse is 2b^two/a.
Transverse Centrality: The line passing through the two foci and the center of the ellipse is chosen the transverse centrality.
Conjugate Axis: The line passing through the center of the ellipse and perpendicular to the transverse axis is chosen the cohabit axis
Eccentricity: (e < 1). The ratio of the distance of the focus from the middle of the ellipse, and the altitude of one end of the ellipse from the centre of the ellipse. If the distance of the focus from the center of the ellipse is 'c' and the altitude of the end of the ellipse from the center is 'a', then eccentricity e = c/a.

Standard Equation of an Ellipse

There are two standard equations of the ellipse. These equations are based on the transverse axis and the conjugate axis of each of the ellipse. The standard equation of the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^ii} = 1\) has the transverse centrality as the x-axis and the conjugate centrality as the y-axis. Further, some other standard equation of the ellipse is \(\dfrac{x^2}{b^2} + \dfrac{y^two}{a^2} = 1\) and it has the transverse axis as the y-axis and its conjugate centrality as the x-axis. The below image shows the two standard forms of equations of an ellipse.
Standard Equations of a Ellipse

Derivation of Ellipse Equation

The first step in the procedure of deriving the equation of the ellipse is to derive the relationship betwixt the semi-major axis, semi-minor axis, and the distance of the focus from the center. The aim is to observe the relationship across a, b, c. The length of the major axis of the ellipse is 2a and the length of the minor centrality of the ellipse is 2b. The distance between the foci is equal to 2c. Allow us take a point P at one end of the major axis and aim at finding the sum of the distances of this betoken from each of the foci F and F'.

PF + PF' = OP - OF + OF' + OP

= a - c + c + a

PF + PF' = 2a
Derivation - Equation of a Ellipse

Now let us accept another point Q at 1 cease of the minor axis and aim at finding the sum of the distances of this point from each of the foci F and F'.

QF + QF' = \(\sqrt{b^2 + c^two}\) + \(\sqrt{b^2 + c^ii}\)

QF + QF' = ii\(\sqrt{b^2 + c^2}\)

The points P and Q prevarication on the ellipse, and as per the definition of the ellipse for any indicate on the ellipse, the sum of the distances from the 2 foci is a abiding value.
2\(\sqrt{b^ii + c^2}\) = 2a

\(\sqrt{b^2 + c^2}\) = a

b² + c² = aⁱⁱ

c^two = a² - bⁱⁱ

Let u.s.a. now check, how to derive the equation of an ellipse. Now we consider any signal S(x, y) on the ellipse and take the sum of its distances from the two foci F and F', which is equal to 2a units. If nosotros detect the higher up few steps, we have already proved that the sum of the distances of whatsoever point on the ellipse from the foci is equal to 2a units.

SF + SF' = 2a

\(\sqrt{(x + c)^2 + y^ii}\) + \(\sqrt{(ten - c)^2 + y^2}\) = 2a

\(\sqrt{(x + c)^2 + y^ii}\) = 2a - \(\sqrt{(x - c)^ii + y^ii}\)

Now we need to square on both sides to solve farther.

(ten + c)² + y^two = 4a²+ (x - c)ⁱⁱ + y² - 4a\(\sqrt{(x - c)^two + y^ii}\)

x² + c^two + 2cx + yⁱⁱ = 4a²+ ten² + c² - 2cx + y² - 4a\(\sqrt{(x - c)^2 + y^2}\)

4cx - 4a^two = - 4a\(\sqrt{(x - c)^ii + y^2}\)

aⁱⁱ - cx = a\(\sqrt{(x - c)^2 + y^2}\)

Squaring on both sides and simplifying, we have.

\(\dfrac{ten^2}{a^2} - \dfrac{y^ii}{c^2 - a^2} =i\)

Since we have c^two = a² - b²nosotros tin substitute this in the to a higher place equation.

\(\dfrac{x^ii}{a^two} + \dfrac{y^two}{b^2} =one\)

This derives the standard equation of the ellipse.

Ellipse Formulas

At that place are dissimilar formulas associated with the shape ellipse. These ellipse formulas tin can be used to calculate the perimeter, area, equation, and other important parameters.

Perimeter of an Ellipse Formulas

Perimeter of an ellipse is defined as the total length of its boundary and is expressed in units like cm, chiliad, ft, yd, etc. The perimeter of ellipse can be approximately calculated using the general formulas given equally,
P ≈ π (a + b)

P ≈ π √[ 2 (aⁱⁱ + b^two) ]

P ≈ π [ (3/2)(a+b) - √(ab) ]

where,

a = length of semi-major axis
b = length of semi-minor axis

Expanse of Ellipse Formula

The area of an ellipse is defined as the full area or region covered by the ellipse in ii dimensions and is expressed in foursquare units like in², cm^two, m², yd², ft², etc. The surface area of an ellipse tin can be calculated with the help of a general formula, given the lengths of the major and small axis. The area of ellipse formula can be given as,

Area of ellipse = π a b
where,

a = length of semi-major axis
b = length of semi-minor axis

Eccentricity of an Ellipse Formula

Eccentricity of an ellise is given as the ratio of the distance of the focus from the center of the ellipse, and the distance of 1 end of the ellipse from the center of the ellipse

Eccentricity of an ellipse formula, e = \( \dfrac ca = \sqrt{1- \dfrac{b^2}{a^2} }\)

Latus Rectum of Ellipse Formula

Latus rectum of of an ellipse tin be defined as the line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The formula to find the length of latus rectum of an ellipse tin can be given as,

L = 2bⁱⁱ/a

Formula for Equation of an Ellipse

The equation of an ellipse formula helps in representing an ellipse in the algebraic form. The formula to find the equation of an ellipse tin can be given every bit,

Equation of the ellipse with centre at (0,0) : x²/a²+ y²/b²= 1

Equation of the ellipse with centre at (h,k) : (x-h)² /a² + (y-m)ⁱⁱ/ b² =1

Example: Find the area of an ellipse whose major and minor axes are xiv in and 8 in respectively.

Solution:

To discover: Area of an ellipse

Given: 2a = fourteen in

a = 14/two = 7

2b = eight in

b = viii/2 = 4

Now, applying the ellipse formula for area:

Area of ellipse = π(a)(b)

= π(seven)(4)

= 28π

= 28(22/7)

= 88 in²

Answer: Expanse of the ellipse = 88 in².

Properties of an Ellipse

At that place are different properties that help in distinguishing an ellipse from other like shapes. These properties of an ellipse are given as,

An ellipse is created by a airplane intersecting a cone at the bending of its base.
All ellipses have 2 foci or focal points. The sum of the distances from any signal on the ellipse to the 2 focal points is a constant value.
There is a centre and a major and small-scale centrality in all ellipses.
The eccentricity value of all ellipses is less than i.

Properties of a Ellipse

Allow usa bank check through three of import terms relating to an ellipse.

Auxilary Circumvolve: A circumvolve fatigued on the major axis of the ellipse is called the auxiliary circle. The equation of the auxiliary circle to the ellipse is x² + yⁱⁱ = a^two.
Manager Circle: The locus of the points of intersection of the perpendicular tangents drawn to the ellipse is called the director circle. The equation of the managing director circle of the ellipse is x^two + y^two = aⁱⁱ + bⁱⁱ
Parametric Coordinates: The parametric coordinates of whatever point on the ellipse is (x, y) = (aCosθ, aSinθ). These coordinates correspond all the points of the coordinate axes and it satisfies all the equations of the ellipse.

How to Draw an Ellipse?

To draw an ellipse in math, there are certain steps to be followed. The stepwise method to draw an ellipse of given dimensions is given below.

Decide what will be the length of the major axis, because the major axis is the longest diameter of an ellipse.
Draw 1 horizontal line of the major centrality' length.
Mark the mid-point with a ruler. This tin be done by taking the length of the major axis and dividing it by two.
Construct a circle of this bore with a compass.
Make up one's mind what will be the length of the small centrality, because the minor axis is the shortest bore of an ellipse.
Now, at the mid-indicate of the major centrality, you take the protractor and gear up its origin. At ninety degrees, mark the point. Then swing 180 degrees with the protractor and mark the spot. You may now draw the small-scale axis between or within the two marks at its midpoint.
Depict a circumvolve of this diameter with a compass as we did for the major axis.
Use a compass to separate the entire circle into twelve xxx degree parts. Setting your protractor on the main axis at the origin and labeling the intervals of 30 degrees with dots volition do this. Then with lines, yous can link the dots through the middle.
Depict horizontal lines (except for the major and minor axes) from the inner circle.
They are parallel to the main centrality, and from all the points where the inner circle and xxx-caste lines converge, they go outward.
Try drawing the lines a trivial shorter near the minor centrality, merely draw them a piffling longer every bit you motility toward the major axis.
Draw vertical lines (except for the major and minor axes) from the outer circle.
These are parallel to the modest axis, and from all the points where the outer circle and 30-degree lines converge, they get inward.
Attempt to draw the lines a lilliputian longer most the minor axis, simply when you lot step towards the main axis, draw them a little shorter.
Y'all tin take a ruler and stretch it a little earlier drawing the vertical line if you detect that the horizontal line is too far.
Do your best with freehand cartoon to depict the curves between the points by hand.

Graph of Ellipse

Let us see the graphical representation of an ellipse with the aid of ellipse formula. At that place are sure steps to be followed to graph ellipse in a cartesian plane.

Step 1: Intersection with the co-ordinate axes

The ellipse intersects the x-axis in the points A (a, 0), A'(-a, 0) and the y-axis in the points B(0,b), B'(0,-b).

Step two : The vertices of the ellipse are A(a, 0), A'(-a, 0), B(0,b), B'(0,-b).

Step 3 : Since the ellipse is symmetric about the coordinate axes, the ellipse has two foci South(ae, 0), S'(-ae, 0) and two directories d and d' whose equations are \(x = \frac{a}{e}\) and \(x = \frac{-a}{e}\). The origin O bisects every chord through it. Therefore, origin O is the heart of the ellipse. Thus it is a key conic.

Step 4: The ellipse is a airtight bend lying entirely within the rectangle bounded by the four lines \(ten = \pm a\) and \(y = \pm b\).

Step 5: The segment \(AA'\) of length \(2a\) is called the major centrality and the segment \(BB'\) of length \(2b\) is called the minor axis. The major and small axes together are called the principal axes of the ellipse.

The length of semi-major axis is \(a\) and semi-small axis is b.

Coordinate Axis - Ellipse

Related Topics:

Coordinate Geometry
Conics in Real Life
Cartesian Coordinates
Parabola
Hyperbola

Examples on Ellipse

Example 1: What are the values of 'a' and 'b' in the equation of ellipse 16x² + 25y² = 1600.

Solution:

To get the values of 'a' and 'b' we need to write the given equation in standard form

So divide the given equation of an ellipse 16x² + 25y² = 1600 by 1600.

we go \(\frac{{x^2}}{100} + \frac{{y^2}}{64} = one\)

So past comparison \(\frac{{x^2}}{100} + \frac{{y^2}}{64} = 1\) with \(\frac{{10^ii}}{{a^2}} + \frac{{y^2}}{{b^2}} = one\)

The values are a = 10 and b = 8.
Case two: Notice the lengths of major and pocket-size axes of the ellipse \(\frac{{10^ii}}{25} + \frac{{y^2}}{sixteen} = i\).

Solution:

The equation of the ellipse is \(\frac{{x^2}}{25} + \frac{{y^ii}}{16} = 1\)

Comparison the above equation with \(\frac{{10^two}}{a^2} + \frac{{y^2}}{b^2} = 1\),

\(a^two=25\) and \(b^2=16\)

Length of major centrality = 2a = 10.

Length of pocket-sized axis = 2b = 8.
Case 3: Find the equation of the ellipse, having major axis along the x-centrality and passing through the points (-iii, 1) and (2, -2).

Solution:

Since the points (-iii, ane) and (2, -2) lie on the ellipse,

\(\frac{{ten^ii}}{a^2} + \frac{{y^2}}{b^2} = 1\), \(\frac{{ix}}{a^2} + \frac{{1}}{b^2} = 1\) and

\(\frac{{four}}{a^2} + \frac{{4}}{b^ii} = 1\)

Solving these equations simultaneously \(a^two = \frac{{32}}{3}\) and \(b^ii = \frac{{32}}{5}\)

So the equation of the ellipse is

\(\frac{{ten^two}}{{\frac{{32}}{3}}} + \frac{{y^2}}{{\frac{{32}}{5}}} = 1\)

i.e. \(3x^2 + 5y^ii = 32\)
Case 4: The length of the semi-major and semi-minor centrality of an ellipse is 5 in and 3 in respectively. Discover its eccentricity and the length of the latus rectum.

Solution:

To find: Eccentricity and the length of the latus rectum of an ellipse.

Given: a = 5 in, and b = 3 in

Now, applying ellipse formula for eccentricity:

\(e = \sqrt{1- \dfrac{ b^ 2}{ a^ 2} }\)

= √( 1 - 3² / 5² )

= √(1 - 9/25 )

= √((25 - 9)/25)

= √(16/25)

= 4/5

= 0.8

Now, applying ellipse formula for latus rectum:

L = 2 b ² /a

= ii(3²)/5

= 2(9)/5

= 18/5

= 3.6 cm

Eccentricity and the length of the latus rectum of the ellipse are 0.eight and iii.6 in respectively.

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Practice Questions on Ellipse

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FAQs on Ellipse

What is Ellipse?

An ellipse is the locus of a point whose sum of the distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse, and the equation of the ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^two}{b^2} = one\). Hither a is chosen the semi-major axis and b is called the semi-minor centrality of the ellipse.

What is the Equation of Ellipse?

The equation of the ellipse is \(\dfrac{x^ii}{a^two} + \dfrac{y^2}{b^2} = 1\). Hither a is called the semi-major centrality and b is the semi-minor centrality. For this equation, the origin is the eye of the ellipse and the x-axis is the transverse axis, and the y-axis is the conjugate axis.

What are the Properties of Ellipse?

The different properties of an ellipse are as given below,

An ellipse is created by a plane intersecting a cone at the angle of its base.
All ellipses have two foci, a heart, and a major and small-scale centrality.
The sum of the distances from any point on the ellipse to the two foci gives a constant value.
The value of eccentricity for all ellipses is less than one.

How to Find Equation of an Ellipse?

The equation of the ellipse tin can be derived from the basic definition of the ellipse: An ellipse is the locus of a point whose sum of the distances from two fixed points is a constant value. Let the stock-still betoken be P(10, y), the foci are F and F'. Then the status is PF + PF' = 2a. This on further substitutions and simplification we have the equation of the ellipse as \(\dfrac{x^2}{a^ii} + \dfrac{y^2}{b^ii} = ane\).

What is the Eccentricity of Ellipse?

The eccentricity of the ellipse refers to the measure out of the curved feature of the ellipse. For an ellipse, the eccentric is always greater than one. (eastward < i). Eccentricity is the ratio of the distance of the focus and ane end of the ellipse, from the center of the ellipse. If the distance of the focus from the center of the ellipse is 'c' and the distance of the stop of the ellipse from the heart is 'a', then eccentricity east = c/a.

What is the General Equation of an Ellipse?

The general equation of ellipse is given equally, \(\dfrac{x^two}{a^ii} + \dfrac{y^two}{b^2} = one\), where, a is length of semi-major axis and b is length of semi-pocket-sized axis.

What are the Foci of an Ellipse?

The ellipse has 2 foci, F and F'. The midpoint of the two foci of the ellipse is the heart of the ellipse. All the measurements of the ellipse are with reference to these two foci of the ellipse. Every bit per the definition of an ellipse, an ellipse includes all the points whose sum of the distances from the two foci is a constant value.

What is the Standard Equation of an Ellipse?

The standard equation of an ellipse is used to correspond a full general ellipse algebraically in its standard grade. The standard equations of an ellipse are given every bit,

\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = one\), for the ellipse having the transverse axis as the 10-axis and the conjugate axis every bit the y-axis.
\(\dfrac{ten^2}{b^2} + \dfrac{y^2}{a^2} = 1\), for the ellipse having transverse centrality as the y-axis and its conjugate centrality as the 10-axis.

What is the Cohabit Axis of an Ellipse?

The axis passing through the middle of the ellipse, and which is perpendicular to the line joining the two foci of the ellipse is called the cohabit axis of the ellipse. For a standard ellipse \(\dfrac{10^2}{a^2} + \dfrac{y^2}{b^2} = i\), its modest axis is y-axis, and information technology is the conjugate axis.

What are Asymptotes of Ellipse?

The ellipse does not have any asymptotes. Asymptotes are the lines drawn parallel to a bend and are causeless to meet the curve at infinity. Nosotros can depict asymptotes for a hyperbola.

What are the Vertices of an Ellipse?

There are 4 vertices of the ellipse. The length of the major centrality of the ellipse is 2a and the endpoints of the major centrality is (a, 0), and (-a, 0). The length of the minor centrality of the ellipse is 2b and the endpoints of the minor axis is (0, b), and (0, -b).

How to Find Transverse Centrality of an Ellipse?

The line passing through the two foci and the eye of the ellipse is called the transverse axis of the ellipse. The major centrality of the ellipse falls on the transverse axis of the ellipse. For an ellipse having the center and the foci on the x-centrality, the transverse axis is the x-axis of the coordinate arrangement.

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Source: https://www.cuemath.com/geometry/ellipse/