HYPERBOLA
A smooth curve on the plane that is defined by the properties of geometry or through equations of which it is the solution set is a hyperbola. It has two parts called connected elements or arms, that are exact duplicates of each other and which are similar to that of two infinite bows. A conic section which is formed by the intersection of a plane and a double cone is a hyperbola. The standard equation of the hyperbola is [(x – h)2 / a2] – [(y – k)2 / b2] = 1.
Cases, where a hyperbola arises, are as follows:
- The curve of a function y = 1 / x in the cartesian plane.
- The path traced by the tip of a sundial.
- The shape of the orbit of a spacecraft while it is gravity assisted.
- The path of a single-apparition comet.
The following are the parts of a hyperbola:
- Foci: The two fixed points.
- Centre: The midpoint of the line segment joining the foci.
- Transverse axis: The line passing through the foci.
- Conjugate axis: The line perpendicular to the transverse axis and passing through the centre.
- Vertices: The points at which the hyperbola intersects the transverse axis.
- Asymptotes: The two imaginary lines that a hyperbola is bounded by.
Properties of hyperbola
A hyperbola is said to be rectangular or equilateral if the transverse and conjugate axes lengths are the same.A rectangular hyperbola has an eccentricity of √2 which is the same as the length of its latus rectum of the axes.There exist two lines which intersect the centre of the hyperbola. The tangents to the centre are called the asymptotes of the hyperbola.The latus rectum of the hyperbola can be defined as the line perpendicular to the transverse axis and passing through any of the foci which are parallel to the conjugate axis. It is given by 2b2 / a.
CIRCLE
A special case of an ellipse is a circle. A geometrical shape in which all the points are equidistant from the centre is a circles. The standard equation of a circle is (x – h)2 + (y – k)2 = r2.
A few examples include a coin, a wheel.
The following are the parts of a circle:
- Diameter: A line passing through the centre that touches two points on the edge of the circle.
- Radius: The distance from the middle or centre of the circle.
- Chord: The line joining two points on the circumference of the circle.
- Tangent: The line which touches the circle at one point.
- Arc: It is a part of the circumference of the circle.
- Sector: The part enclosed by two radii of a circle and their intercepted arc.
- Segment: A chord that divides the circle into two regions.
Properties of circle
The circles with equal radii are congruent to each other.the longest chord of the circle is the diameter. Its distance from the centre of the circle is 0.Identical chords and identical circles possess identical circumference. The chord is bisected by the radius which is drawn perpendicular to the chord. The circles are said to be similar if they have different radii. A rectangle, trapezium, triangle, square, kite can be circumscribed by a circle. A square, triangle and kite can inscribe a circle inside them.The chords are equal in length if they are equidistant from the centre. The tangents are parallel to each other if they are drawn at the end of the diameter.
