Parabola can be defined as the conic section created by Intersection of a plane which is parallel to the generator line of surface and a Right Circular Cone. The other definition of parabola: A Plane Curve obtained by the moving a Point which is at fixed distance from a fixed point and also equals to the distance form a fixed line is known as parabola. Now we will see how to draw a parabola.
For drawing parabola we need to follow some steps which are given below:
Step 1: First we take a ‘diretrix’ or a Line Segment.
Step 2: Then we draw focal point named as ‘F’.
Step 3: If we want the vertex then we bisect the distance between line P and focal point ‘F’.
Step 4: Then we draw vertical lines of any distance and name them A, B, C, D, E.
Step 5: Then we measure the distance from Line Segment ‘P’ to the point ‘A’.
Step 6: Then we measure the distance from line segment to the point ‘B’ and repeat the same procedure till the last point.
Step 7: We draw the point from the line segment.
Step 8: At last we join all the points and we get parabola.
Now we will see the general form of parabola:
The general form is:
=> (αp + βq)2 + γp + δq + ∈= 0;
This above equation is obtained from the general conic equation which is given below:
=> Ap2+ Bpq + Cq2 + Dp + Eq + F = 0;
And the equations for a general form of parabola maths with the focus point F(s, t) and a directrix in the form:
=> pa + qy + c = 0;
=> (pa + qy + c)2 = (p - s)2 + (q - t)2
P2 + q2
This is the equation of parabola in maths.
If the Intersection of a Right Circular Cone and a plane that is also parallel to an element of the cone and the locus points of cone is equidistance from a fixed Point then we get a Plane Curve and that curve is said to be a Parabola. And in other words we can say that in mathematical Geometry a parabola is a curve that is obtained by rounding a point which is positioned at fixed distance from a fixed point and is also equal to the distance form a fixed line is said to be a parabola. And parabola is also follows the properties of a Hyperbola.
Let’s see how to graph a parabola.
To graph parabola we have to follow some steps so that it is easier to understand:
Step1: To graph a parabola first we have to take a Line Segment.
Step2: Then after draw focal point and named as F for the construction of parabola.
Step3: If we want to find the vertex of a line then bisect the distance between line and focal point F.
Step4: And if we want to find the vertex of a parabola then we have to bisect the distance which is measure in between line and focal point.
Step5: Draw vertical lines of any distance apart.
Step6: Measure the distance from Line Segment to the point.
Step7: Find the distance from line segment to the other point and repeat the same procedure till the last point.
Step8: we draw point from the line segment.
Step9: At last we have to join all the points and we get parabola.
Now we will see the Equation of Parabola.
The Equation for a Parabola on vertical axis is given by:
(x – s)2 = 4p (y – t); where ‘s’ and ‘t’ are the coordinates of the vertex and ‘p’ is the distance from the vertex to the focus and the vertex to the directrix.
This is the Equation of a Parabola.
Equation such as y = ax2
or x = by2
represent a Parabola. Graph will incline towards higher power variable. Parabola can be defined as the Set of all those points in a plane which are at away by same distance from a fixed Point and fixed line.
This fixed point, in this case, is known as the focus (F)and fixed line (L) is called directrix of parabola. Line 'L' in plane does not contain focus (F). Directrix is actually normal (perpendicular) to Axis of Symmetry. If x-axis is the symmetric axis then y-axis will be directrix and vice versa. In short, we can say that if axis of symmetry is horizontal then directrix of parabola will be a vertical line and vice versa. Normally directrix is the line which does not touch parabolic graph.
Following parabolic graph can be taken into account to understand the directrix of a parabola.
Red line in above graph is representing the axis of symmetry which divides the parabolic graph into two identical graphs. Perpendicular line to the axis of symmetry is known as directrix. It is shown in purple color. Green dot is pointing out the focus and orange dot is indicating the vertex. Axis of symmetry is passing through the focus and vertex of given parabolic graph and is normal to the directrix.
Vertex of parabola can be defined as the point which is located at midpoint of focus and directrix that is the vertex is situated at equal distance from directrix of parabola and focus between them. Orange dot in above figure is located at mid of the locus and parabola directrix.
Word Parabola was discovered by 'Apollonius', while he was studying conic sections. Parabola is a part of conic section which defines concept of Geometry. It can also be defined as fixed points which are equidistant from fixed Point in the plane in which it is drawn. Fixed line on a plane in which we draw a parabola is called as directrix and fixed point in the plane is called focus. Line which is perpendicular to directrix and which passes through the focus of the parabola is known as the line of Symmetry. The point on the symmetry axis which intersects on the parabola is known as the vertex, it is basically the point where curvature of the parabola is greatest. So the definition of parabola vertex is the point where the curvature of the parabola is greatest than on any other point on parabola. It is as shown in the diagram. Some of the properties of the parabola are:
1. Axis of parabola: the line perpendicular to the directrix and passing through focus of parabola is known as axis of parabola.
2. Focus of parabola: It is the point in inner side of parabola having x- coordinate as zero.
3. Vertex of a parabola: It is the point on symmetric axis, where curvature of parabola is greatest.
Equation of parabola: y2= 4 ax in positive x –axis, y2= -4ax on negative x- axis, x2 = 4ay for positive y- axis and x2 = -4ay for negative y- axis.
Where 'y' is the y- coordinate and 'x' is x- coordinate.
Diagram showing parabola and various properties like: vertex, axis of symmetry, directrix and focus is shown below:
Diagram of a parabola: