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An object falling under the effect of gravity after being thrown upwards, the projection of a circle, and the curvilinear physical motion of bodies have one thing in common. When you trace the path of a projectile falling under the effect of gravity alone and then look at the projection of a circle, you will find that both of them have a similar shape. This shape is known as a standard parabola.
Many great scientists, like Galileo Galilei and Pascal, have talked about the importance of the standard parabola shape and the different phenomena in which it occurs. In mathematics, a shape on the coordinate plane that is mirror-symmetrical and usually is in the shape of the letter U is called a parabola.
Definition
A parabola is a locus of points that fulfil certain conditions. For example, a circle can be described as a locus of points that are equidistant to a fixed point in space. This point is called the centre of the circle.
Similarly, a parabola is an intersection of a conic section and defined as the locus of all points at an equal distance from a given point and line in space. A parabola can also be defined as the equation of a curve such that any point that satisfies the equation of the curve will be equidistant to a particular point and line in space.
A parabola is an important shape in the conic sections that form a part of coordinate geometry. Many real-life applications are present in the shape of the parabola. An example of this is the interior of a car’s headlight. It has a parabolic shape to reflect more light towards the front. Another example is the head of a space shuttle or a ballistic missile. It is usually in the form of a parabola to avoid huge aerodynamic resistance while keeping the amount of pressure on the top in check.
General equation of a parabola
A parabola can be drawn in two ways. The first way is to draw it such that the open mouth of the parabola is towards the positive or negative y-axis. The second way is to draw a parabola such that the open mouth of the parabola is towards the positive or negative x-axis.
If the parabola is drawn in a manner such that the open mouth faces the y-axis, then it is called a regular parabola. Its equation is given as:
y=a(x-h)2+k
If the parabola is drawn with its open mouth towards the x-axis, it is said to be a sideways parabola. Its equation is given as:
x=a(y-k)2+h
Standard equation of a parabola
A standard parabola can be defined as a parabola in which the vertex is the origin. Hence, the vertex (h,k) becomes (0,0). The change in the vertex of the parabola results in the standard equation:
y2=4ax
There are four standard parabolas in coordinate geometry. They differ in their conjugate and transverse axes. These standard forms are created with respect to the axis and orientation of each parabola.
When a parabola has its vertex on the origin, is symmetric about the x-axis, and opens towards the right side of the coordinate plane, its equation is given as:
y2=4ax
When a parabola has its vertex on the origin, is symmetric about the x-axis, and opens towards the left side of the coordinate plane, its equation is given as:
y2=-4ax
When a parabola has its vertex on the origin, is symmetric about the y-axis, and opens upwards on the coordinate plane, its equation is given as:
x2=4ay
When a parabola has its vertex on the origin, is symmetric about the y-axis, and opens downwards on the coordinate plane, its equation is given as:
x2=-4ay
Here, we can observe that if the parabola is symmetric about the x-axis, then the squared term in the standard parabola equation is the y variable. On the contrary, if the parabola is symmetric on the y-axis, then the squared term in the equation of the parabola is the x variable.
Another observation that can be made is if the parabola is symmetric about the x-axis, then the sign of the RHS in the equation of the parabola is given by the side towards which the parabola opens. If the parabola opens towards the left side, the sign is negative; if it opens towards the right, the sign is positive.
Similarly, if the parabola is symmetric about the y-axis, the sign of the RHS in the equation of the parabola is governed by the direction in which the parabola opens. If it opens in the upwards direction, the sign is positive; if it opens in the downward direction, the sign is negative.
Conclusion
A parabola is an intersection of a conic section and is defined as the locus of all points at an equal distance from a given point and line in space. It can also be defined as the equation of a curve such that any point that satisfies the equation of the curve will be equidistant to a particular point and line in space.
A standard parabola can be defined as one in which the vertex of the parabola is the origin. Therefore, the value of the vertex (h,k) becomes (0,0). The change in the vertex of the parabola results in the standard equation of the parabola to become:
y2=4ax
