Ellipse is an integral part of the conic section and is similar in properties to a circle. Unlike the circle, an ellipse is oval in shape. An ellipse has an eccentricity less than one, and it represents the locus of points, the sum of whose distances from the two foci of the ellipse is a constant 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. Show
Here we shall aim at knowing the definition of an ellipse, the derivation of the equation of an ellipse, and the different standard forms of equations of the ellipse. What is an Ellipse?An ellipse in math is the locus of points in a plane in such a way that their distance from a fixed point has a constant ratio of 'e' to its distance from a fixed line (less than 1). 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 called the focus and is denoted by S, the constant ratio 'e' as the eccentricity, and the fixed line is called as directrix (d) of the ellipse. Ellipse DefinitionAn ellipse is the locus of points in a plane, the sum of whose distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse. Ellipse EquationThe general equation of an ellipse is used to algebraically represent an ellipse in the coordinate plane. The equation of an ellipse can be given as, \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) Parts of an EllipseLet us go through a few important terms relating to different parts of an ellipse.
Standard Equation of an EllipseThere 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 \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) has the transverse axis as the x-axis and the conjugate axis as the y-axis. Further, another standard equation of the ellipse is \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\) and it has the transverse axis as the y-axis and its conjugate axis as the x-axis. The below image shows the two standard forms of equations of an ellipse. Derivation of Ellipse EquationThe first step in the process of deriving the equation of the ellipse is to derive the relationship between the semi-major axis, semi-minor axis, and the distance of the focus from the center. The aim is to find the relationship across a, b, c. The length of the major axis of the ellipse is 2a and the length of the minor axis of the ellipse is 2b. The distance between the foci is equal to 2c. Let us take a point P at one end of the major axis and aim at finding the sum of the distances of this point from each of the foci F and F'. PF + PF' = OP - OF + OF' + OP = a - c + c + a PF + PF' = 2a Now let us take another point Q at one end 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' = \(\sqrt{b^2 + c^2}\) + \(\sqrt{b^2 + c^2}\) QF + QF' = 2\(\sqrt{b^2 + c^2}\) The points P and Q lie on the ellipse, and as per the definition of the ellipse for any point on the ellipse, the sum of the distances from the two foci is a constant value. \(\sqrt{b^2 + c^2}\) = a b2 + c2 = a2 c2 = a2 - b2 Let us now check, how to derive the equation of an ellipse. Now we consider any point 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 we observe the above few steps, we have already proved that the sum of the distances of any point on the ellipse from the foci is equal to 2a units. SF + SF' = 2a \(\sqrt{(x + c)^2 + y^2}\) + \(\sqrt{(x - c)^2 + y^2}\) = 2a \(\sqrt{(x + c)^2 + y^2}\) = 2a - \(\sqrt{(x - c)^2 + y^2}\) Now we need to square on both sides to solve further. (x + c)2 + y2 = 4a2 + (x - c)2 + y2 - 4a\(\sqrt{(x - c)^2 + y^2}\) x2 + c2 + 2cx + y2 = 4a2 + x2 + c2 - 2cx + y2 - 4a\(\sqrt{(x - c)^2 + y^2}\) 4cx - 4a2 = - 4a\(\sqrt{(x - c)^2 + y^2}\) a2 - cx = a\(\sqrt{(x - c)^2 + y^2}\) Squaring on both sides and simplifying, we have. \(\dfrac{x^2}{a^2} - \dfrac{y^2}{c^2 - a^2} =1\) Since we have c2 = a2 - b2 we can substitute this in the above equation. \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} =1\) This derives the standard equation of the ellipse. Ellipse FormulasThere are different formulas associated with the shape ellipse. These ellipse formulas can be used to calculate the perimeter, area, equation, and other important parameters. Perimeter of an Ellipse FormulasPerimeter of an ellipse is defined as the total length of its boundary and is expressed in units like cm, m, ft, yd, etc. The perimeter of ellipse can be approximately calculated using the general formulas given as, P ≈ π √[ 2 (a2 + b2) ] P ≈ π [ (3/2)(a+b) - √(ab) ] where,
Area of Ellipse FormulaThe area of an ellipse is defined as the total area or region covered by the ellipse in two dimensions and is expressed in square units like in2, cm2, m2, yd2, ft2, etc. The area of an ellipse can be calculated with the help of a general formula, given the lengths of the major and minor axis. The area of ellipse formula can be given as, Area of ellipse = π a b
Eccentricity of an Ellipse FormulaEccentricity of an ellise is given as the ratio of the distance of the focus from the center of the ellipse, and the distance of one 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 FormulaLatus rectum of of an ellipse can 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 can be given as, L = 2b2/a Formula for Equation of an EllipseThe equation of an ellipse formula helps in representing an ellipse in the algebraic form. The formula to find the equation of an ellipse can be given as, 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-k)2/ b2 =1 Example: Find the area of an ellipse whose major and minor axes are 14 in and 8 in respectively. Solution: To find: Area of an ellipse Given: 2a = 14 in a = 14/2 = 7 2b = 8 in b = 8/2 = 4 Now, applying the ellipse formula for area: Area of ellipse = π(a)(b) = π(7)(4) = 28π = 28(22/7) = 88 in2 Answer: Area of the ellipse = 88 in2. Properties of an EllipseThere are different properties that help in distinguishing an ellipse from other similar shapes. These properties of an ellipse are given as,
Let us check through three important terms relating to an 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.
Graph of EllipseLet us see the graphical representation of an ellipse with the help of ellipse formula. There are certain 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 2 : 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 S(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 centre of the ellipse. Thus it is a central conic. Step 4: The ellipse is a closed curve lying entirely within the rectangle bounded by the four lines \(x = \pm a\) and \(y = \pm b\). Step 5: The segment \(AA'\) of length \(2a\) is called the major axis and the segment \(BB'\) of length \(2b\) is called the minor axis. The major and minor axes together are called the principal axes of the ellipse. The length of semi-major axis is \(a\) and semi-minor axis is b. Related Topics:
go to slidego to slidego to slidego to slide
Breakdown tough concepts through simple visuals. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. Book a Free Trial Class
FAQs on EllipseAn 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 \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). Here a is called the semi-major axis and b is called the semi-minor axis of the ellipse. What is the Equation of Ellipse?The equation of the ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). Here a is called the semi-major axis and b is the semi-minor axis. For this equation, the origin is the center 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,
How to Find Equation of an Ellipse?The equation of the ellipse 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 fixed point be P(x, y), the foci are F and F'. Then the condition is PF + PF' = 2a. This on further substitutions and simplification we have the equation of the ellipse as \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). What is the Eccentricity of Ellipse?The eccentricity of the ellipse refers to the measure of the curved feature of the ellipse. For an ellipse, the eccentric is always greater than one. (e < 1). Eccentricity is the ratio of the distance of the focus and one 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 end of the ellipse from the center is 'a', then eccentricity e = c/a. What is the General Equation of an Ellipse?The general equation of ellipse is given as, \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), where, a is length of semi-major axis and b is length of semi-minor axis. What are the Foci of an Ellipse?The ellipse has two foci, F and F'. The midpoint of the two foci of the ellipse is the center of the ellipse. All the measurements of the ellipse are with reference to these two foci of the ellipse. As 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 represent a general ellipse algebraically in its standard form. The standard equations of an ellipse are given as,
What is the Conjugate Axis of an Ellipse?The axis passing through the center of the ellipse, and which is perpendicular to the line joining the two foci of the ellipse is called the conjugate axis of the ellipse. For a standard ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), its minor axis is y-axis, and it is the conjugate axis. What are Asymptotes of Ellipse?The ellipse does not have any asymptotes. Asymptotes are the lines drawn parallel to a curve and are assumed to meet the curve at infinity. We can draw asymptotes for a hyperbola. What are the Vertices of an Ellipse?There are four vertices of the ellipse. The length of the major axis of the ellipse is 2a and the endpoints of the major axis is (a, 0), and (-a, 0). The length of the minor axis of the ellipse is 2b and the endpoints of the minor axis is (0, b), and (0, -b). How to Find Transverse Axis of an Ellipse?The line passing through the two foci and the center of the ellipse is called the transverse axis of the ellipse. The major axis of the ellipse falls on the transverse axis of the ellipse. For an ellipse having the center and the foci on the x-axis, the transverse axis is the x-axis of the coordinate system. |