There are so many important topics in geometry and the conic section is one among them.Conic sections are divided into many types based on the angle which is formed between the plane and the intersection of the right circular cone with it.Based on the position of the plane which intersects the angle of the intersection and the cone, four types of conic sections are derived. Hyperbola is one of them.Hyperbola is an open curve that has two branches and they are the mirror image of each other. The two curves are like infinite bows.A hyperbola contains two vertices and two foci. Hyperbola is also known as the mirror image of the parabola.In this article, we will learn about the definition of hyperbola, the properties of hyperbola,the terms related to it and some hyperbola formulas.
Definition:
The hyperbola is a locus of points which are placed in such a way that the distance to each focus is a constant greater than 1. When a plane is intersected by the right circular cone in such a way that the angle between the plane and the vertical axis is less than the vertical angle, a hyperbola is formed. In hyperbola, the locus of the point moving in a plane in such a way that the ratio of its distance from a fixed point, which is known as focus, to that from a fixed line(directrix) is a constant greater than one.
Standard equation of hyperbola:
When the centre of the hyperbola is at the origin and the foci are either on the a- axis or b-axis the equation of the hyperbola is simplest. The standard equation of a hyperbola is:
a2x2-b2y2=1
Wherey2= x2e2- 1
Here “x” is the distance between the centre and the vertex.
Terms related to hyperbola:
- Centre: The midpoint of the line which joins the two foci is the centre of the hyperbola. It is the intersection point of the transverse axis and the conjugate axis.
- Foci: The two points which define the hyperbola are called foci together. They are also known as focus.
- Transverse axis: Transverse axis is the line which is passing through the foci. It is also known as the major axis of hyperbola.
- Conjugate axis: Conjugate axis is the line which is perpendicular to the transverse axis. This line is passing through the centre of the hyperbola.
- Vertex:The vertices of the hyperbola are the two points which intersect the transverse axis.
Properties of hyperbola:
- The difference between the focal distances on the hyperbola is constant. It is the same as the transverse axis.||PS –PS’||= 2a.
- If e1 and e2 represents the eccentricities of hyperbola then the relation e1-2+ e2-2=1 is good.
- If in a hyperbola, the lengths of the transverse axis and the conjugate axis are the same, then the hyperbola will be rectangular or equilateral.
- The line perpendicular to the transverse axis and passing through any one of the foci and it is parallel to the conjugate axis is the latest rectum of the hyperbola. It is 2b²/a.
- The centre of the hyperbola is intersected by two lines. Asymptotes of the hyperbola are the tangents of the centre.
- If the point(a1,b1) lies within, on or outside of the hyperbola the value of a1²/x² — b1²/y²= 1 is positive, zero or negative.
- If a hyperbola is rectangular, the eccentricity of √2 is equal to the length of its latus rectum of the axis.
Some Important formulas of hyperbola: There are some formulas to find out the value of different aspects of hyperbola.
Eccentricity:e= √1+ b²/a². Here he is eccentric.
Vertex: (a, y0) and (-a, y0).
Asymptotes: y= y0- (b/a)x +(b/a)x0.
Y= y0 + (b/a)x – (b/a)x0.
Foci: (x0+√a²+b², y0), and (x0-√a²+b², y0)
Semi-latus rectum(p): p=b²/a
Here, x0,y0 are the centre points.
a= semi-major axis.
b= semi- minor axis.
Though hyperbola is known as the mirror image of parabola, there are so many differences between hyperbola and parabola.To know more about parabola and many other geometry related topics follow the Cuemath app.
So, here we have discussed the hyperbola. To know more about other conic sections and other mathematical topics, download the Cuemath app. This app will help you to learn more and practice various mathematical problems.
