In class-12th, in this chapter you are going to learn about the parabola ,their standard equations, and all the related concepts. You will be studying conic sections which are linked to this topic , where as the name suggests, have something to do with sections of a cone. Different ways of cutting a cone using a plane will give rise to different conics in the parabola.
What is the parabola
In the Section of the right circular cone by a plane parallel to a generator of the cone is a parabola. It is a locus of a point, which moves so that distance from a fixed point (focus) is equal to the distance from a fixed line (directrix)
- Fixed point is called focus
- Fixed line is called directrix
Equation of Parabola Given Focus and Directrix find it
If we consider only parabolas that open upwards or downwards, then the directrix will be a horizontal line of the form y = cy = c .
Let (a,b) (a,b) be the focus and let y = cy = c be the directrix. Let (x0,y0) (x0,y0) be any point on the parabola.
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Any point, (x0,y0) on the parabola satisfies the definition of parabola, so there are two distances to calculate:
The distance between the point on parabola to the focus
The distance between the point on parabola to the directrix
To find the equation of the parabola, equate these two expressions and solve for y0y0 .
Find the equation of the parabola in the example above.
Distance between the point
(x0,y0) and (a,b)
(xo-a)2+(yo-b2
Distance between point
(x0,y0)(x0,y0) and the line y=c:
| y0 − c |
(Here, the distance between the point and horizontal line is the difference of their y-coordinates.)Equate the two expressions.
Equate the two expressions.
xo-a)2+(yo-b)2=Yo-c
Square both sides. (x0−a)2+ (y0−b)2= (y0−c)2
Expand the expression in y0 on the both sides and simplify.
(x0−a)2+b2− c2= 2 (b−c) y0
This equation in (x0,y0) is true for all other values on the parabola and hence we can rewrite with (x,y)(x,y) .
Therefore, the equation of the parabola with focus (a,b) and the directrix y=c is
(x−a)2+b2−c2= 2(b−c)y
Standard Equation
The simplest equation of a parabola is the y2 = x when the directrix is parallel to the y-axis. In the general, if the directrix is parallel to the y-axis in the standard equation of a parabola is given as:
Four Common Forms of a Parabola
| Form: | y2 = 4ax | y2 = – 4ax | x2 = 4ay | x2 = – 4ay |
| Vertex: | (0, 0) | (0,0) | (0, 0) | (0, 0) |
| Focus: | (a, 0) | (-a, 0) | (0, a) | (0, -a) |
| Equation of the directrix: | x = – a | x = a | y = – a | y = a |
| Equation of the axis: | y = 0 | y = 0 | x = 0 | x = 0 |
| Tangent at the vertex: | x = 0 | x = 0 | y = 0 | y = 0 |
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Conclusion:
Congratulations! You have completed the parabola. I hope that you have learned how to graph a parabola. we were introduced to some new concepts like vertex, an axis of symmetry. Also, there was a connection between parabolas and factoring; all these are given in the textbook. “You will find a real-life connection. All these concepts.