Parabola and Hyperbola

When we talk about a parabola, we are referring to an equation of a curve that is equidistant from both a fixed point and a fixed line, and that is defined as follows: The fixed point of the parabola is referred to as the focus, and the fixed line of the parabola is referred to as the directrix of the parabola. Also, it is important to note that the fixed point does not coincide with the fixed line in this case. Any point with a locus that is equal distance from a given point (focus) and an equal distance from a given line (directrix) is referred to as a parabola. The parabola is a significant curve in the conic sections of coordinate geometry, and it has many applications.

Equation of a Parabola

An equilateral parabola has the general equation y = a(x-h)2 + k or x = a(y-k)2 +h, where (h,k) denotes the vertex of the curve. It is standard equation for a regular parabola to have the equation y2 = 4ax.

Some of the key terms listed below will assist you in comprehending the characteristics and components of a parabola.

• The parabola’s focus is located at the point (a, 0).
• The directrix of a parabola is a line drawn parallel to the y-axis and passing through the point (-a, 0). The directrix of the parabola is perpendicular to the axis of the curve.
• The focal chord of a parabola is defined as the chord that passes through the centre of the parabola. The focal chord cuts the parabola at two distinct points, which are indicated by the arrows.
• Distance between the two points of focus: The focal distance is the distance between a point ( x 1, y 1 ) on the parabola and the centre of the parabola. The focal distance is also the same as the perpendicular distance between this point and the directrix, so they are equal.
• In a parabola, the Latus Rectum is the focal chord that is perpendicular to the axis of rotation and passes through the centre of the parabola’s centre of rotation. The length of the latus rectum is calculated using the formula LL’ = 4a. The latus rectum’s endpoints are (a, 2a), (b, 2b), and (c, 2c) (a, -2a).
• Eccentricity is defined as (e = 1). In other words, it is the ratio of the distance between a point and the focus to the distance between a point and the directrix. One unit of eccentricity is equal to one unit of a parabola.

Formula for the Parabola

For example, the Parabola Formula can be used to represent the general shape of a parabolic path in the plane. The formulas that are used to obtain the parameters of a parabola are listed below.

• It is the value of the parameter a that determines the direction of the parabola.
• Vertex = (h,k), where h = -b/2a and k = f(h)
• The Latus Rectum is represented by the number 4a.
• (h, k+ (1/4a)) is the focus
• y = k – 1/4a is the directrix.

Hyperbola

Known as connected components or branches, hyperbolas are a type of smooth curve that lies in a plane and has two pieces that are mirror images of each other and resemble two infinite bows. A hyperbola is composed of two pieces, known as connected components or branches, that are mirror images of each other and resemble two infinite bows. A hyperbola is a collection of points whose difference in distances between two foci has a constant value when viewed from two different directions. This difference is calculated by subtracting the distance from the farther focus from the distance from the nearer focus, and then dividing the result by two. It can be shown that the hyperbola has two foci, P(x,y), and the locus of the hyperbola is PF – PF’ = 2a when there is a point P(x,y) on the hyperbola and two foci F, F’.

Hyperbola are made up of different parts

Let’s go over some of the most important terms that relate to the different parameters of a hyperbola one by one.

• The hyperbola has two foci, and their coordinates are  F(c, o) (-c, 0).

• The centre of the hyperbola is the midpoint of the line connecting the two foci, which is also known as the hyperbola’s centre.

• The length of the major axis of the hyperbola is 2a units.

• Minor Axis: The length of the minor axis of the hyperbola is 2b units.

• The vertices of a hyperbola are the points at which the axis of the hyperbola intersects the axis. The hyperbola’s vertices are (a, 0), and (-a, 0).

• A line drawn perpendicular to the transverse axis of a hyperbola and passing through the foci of the hyperbola is known as the latus rectum of the hyperbola b The hyperbola’s latus rectum has a length of 2b2/a, which is its radius.

• This is referred to as the transverse axis of the hyperbola because it passes through the two foci as well as through its centre of rotation.

• It is referred to as the conjugate axis of a hyperbola when a line passing through the centre of the hyperbola and perpendicular to the transverse axis is parallel to the transverse axis.

• The eccentricity of a hyperbola is defined as (e > 1). The eccentricity of a hyperbola is defined as the relationship between the distance between the focus and the centre of the hyperbola and the distance between the vertex and the centre of the hyperbola. The distance between the focus and the vertex is ‘c’ units, and the distance between the focus and the vertex is ‘a’ units, so the eccentricity is equal to e = c/a.

Equation of the Hyperbola

The general equation of a hyperbola is represented by the equation shown below. Where the x-axis represents the transverse axis of the hyperbola, and the y-axis represents the conjugate axis of the hyperbola, the hyperbola is defined as

Conclusion

Parabola, refers to an equation of a curve that is equidistant from both a fixed point and a fixed line, and that is defined as follows: The fixed point of the parabola is referred to as the focus, and the fixed line of the parabola is referred to as the directrix of the parabola. An equilateral parabola has the general equation y = a(x-h)2 + k or x = a(y-k)2 +h, where (h,k) denotes the vertex of the curve. It is standard equation for a regular parabola to have the equation y2 = 4ax. Hyperbolas are a type of smooth curve that lies in a plane and has two pieces that are mirror images of each other and resemble two infinite bows. A hyperbola is composed of two pieces, known as connected components or branches, that are mirror images of each other and resemble two infinite bows.The hyperbola has two foci, and their coordinates are  F(c, o) (-c, 0).The centre of the hyperbola is the midpoint of the line connecting the two foci, which is also known as the hyperbola’s centre.The length of the major axis of the hyperbola is 2a units.Minor Axis: The length of the minor axis of the hyperbola is 2b units.