Hey there! We receieved your request. The conics like circle, parabola, ellipse and hyperbola are all interrelated and therefore it is crucial to know their distinguishing features as well as similarities in order to attempt the questions in various competitive exams like the JEE.
Hence, students are advised to study these topics attentively in order to remain competitive in the JEE. In this section, we shall discuss the similarities and differences between ellipse and hyperbola in detail. Hence, the equations are closely related, the only difference being of a negative sign. Both ellipses as well as hyperbolas have vertices, foci, and a center. In standard form, both the curves have 0, 0 as the centre. The situation is quite different in case of a hyperbola.
The order of the terms x 2 and y 2 decide whether the transverse axis would be horizontal or vertical. If x 2 comes first then the transverse axis would be horizontal. Conversely, if the y 2 term is listed first, the transverse axis would be vertical. Going into the intricacies, we see that the focal distance in hyperbola is greater than the distance of the vertex. The situation is quite different in case of ellipse; the distance from the centre to vertex is greater than the focal distance.
Hence, again the only difference is of negative b 2. Equation of tangent in slope form:. Therefore, the height of the cable feet from the center of the bridge is 25 feet. Can you see this in the drawing? Lay the two distances down flat. For vertical ellipses , see the table below. You also may have to complete the square to be able to graph an ellipse, like we did here for a circle. We need to first complete the square so we can get the equation in ellipse form.
After you complete the square, divide all terms by 4 , so we have a 1 on the right. You may be asked to write an equation from either a graph or a description of an ellipse:. Write the equation of the ellipse :.
We can see that the ellipse is 10 across the major axis length and 4 down the minor axis length. The foci of ellipses are very useful in science for their reflective properties sound waves, light rays and shockwaves, as examples , and are even used in medical applications. Two girls are standing in a whispering gallery that is shaped like semi-elliptical arch. The height of the arch is 30 feet, and the width is feet. How far from the center of the room should whispering dishes be placed so that the girls can whisper to each other?
Whispering dishes are places at the foci of an ellipse. Each girl would stand 40 feet from the center of the room. An ice rink is in the shape of an ellipse, and is feet long and 75 feet wide. What is the width of the rink 15 feet from a vertex? Note that we need to take double You also may have to complete the square to be able to graph an hyperbola, like we did here for a circle.
We need to first complete the square so we can get the equation in hyperbola form. You may be asked to write an equation from either a graph or a description of a hyperbola, as in the following problem:. So far then we have:.
Like ellipses, the foci of hyperbolas are very useful in science for their reflective properties , and hyperbolic properties are often used in telescopes. Both hyperbola and ellipse are conic sections, and their differences are easily compared in this context.
When the intersection of the conic surface and the plane surface produces a closed curve, it is known as an ellipse. It can also be defined as the locus of the set of points on a plane such that the sum of the distances to the point from two fixed points remains constant. Remember; in elementary math classes the ellipses are drawn using a string tied to two fixed pins, or a string loop and two pins. The line segment passing through the foci is known as the major axis, and the axis perpendicular to the major axis and passing through the center of the ellipse is known as the minor axis.
The diameters along each axis are known as the transverse diameter and the conjugate diameter respectively. Half of the major axis is known as the semi-major axis, and half of the minor axis is known as the semi-minor axis. Eccentricity e is defined as the ratio between the distance from a focus to the arbitrary point PF 2 and the perpendicular distance to the arbitrary point from the directrix PD.
The general equation of the ellipse, when the semi-major axis and the semi-minor axis coincide with the Cartesian axes, is given as follows. In other words, it is a point about which rays reflected from the curve converge. A parabola has one focus about which the shape is constructed; an ellipse and hyperbola have two. A directrix is a line used to construct and define a conic section.
The distance of a directrix from a point on the conic section has a constant ratio to the distance from that point to the focus. As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.
These properties that the conic sections share are often presented as the following definition, which will be developed further in the following section. These distances are displayed as orange lines for each conic section in the following diagram. Parts of conic sections : The three conic sections with foci and directrices labeled. A parabola is the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix.
The point halfway between the focus and the directrix is called the vertex of the parabola. In the next figure, four parabolas are graphed as they appear on the coordinate plane. They may open up, down, to the left, or to the right. Four parabolas, opening in various directions: The vertex lies at the midpoint between the directrix and the focus.
An ellipse is the set of all points for which the sum of the distances from two fixed points the foci is constant. In the case of an ellipse, there are two foci, and two directrices. Ellipse: The sum of the distances from any point on the ellipse to the foci is constant. A hyperbola is the set of all points where the difference between their distances from two fixed points the foci is constant. In the case of a hyperbola, there are two foci and two directrices.
Hyperbolas also have two asymptotes. Hyperbola: The difference of the distances from any point on the ellipse to the foci is constant. The transverse axis is also called the major axis, and the conjugate axis is also called the minor axis. Conic sections are used in many fields of study, particularly to describe shapes. For example, they are used in astronomy to describe the shapes of the orbits of objects in space.
They could follow ellipses, parabolas, or hyperbolas, depending on their properties. It can be thought of as a measure of how much the conic section deviates from being circular. The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix.
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