Concepts of Hyperbola Parabola Ellipse and Circle. Class 11 Maths. JEE MAIN, NEET.
Concepts of Hyperbola Parabola Ellipse and Circle. Class 11 Maths.
Hyperbola
Parabola
Ellipse
Circle
Do you know how they originate? And how important they are in planetary motions?
In this 3 d animated tutorial, you will learn all these answers and concepts.
Hyperbola, parabola, ellipse circle et cetera are based on a same concept. But with slightly different parametric values.
To define any one of them, you need to consider three basic things.
Number one. A fixed straight line. Which is called directrix, in this context.
Number two. A fixed point. It is called focal point. And is named as S
Number three. A moving point. It is named as P.
Now, if p moves in such a manner that, at each and every position, the ratio of the distance from the point p to the focus S, that is S P, and distance from the directrix, that is p m, remains fixed. And also greater than one. Then the trajectory of the point p is called a hyperbola.
The ratio s p by pm is termed as the eccentricity of the conic.
For hyperbola, values of its eccentricity lies between one and infinity.
If the point p moves through a closer path to the directrix, then it becomes flatter. And eventually becoming a straight line, as it touches the directrix.
In this special occasion, eccentricity reaches an infinite value. Since s p is zero in this case.
Isn't interesting?
Thus, straight line is a conic. And its eccentricity is infinity. An infinite eccentricity can be achieved, if the point p, moves through the directrix.
If the point p moves through a closer path to focus, the trajectory tends to become a parabola. We will discuss parabola later in this tutorial.
Actually, hyperbola, by default generated in pair. This second curve is called a conjugate hyperbola. And is a mirror symmetrical to the first curve.
Also, note that, there exist a secondary directrix and a secondary focal point.
The closest distance between these two curves is known as semi major axis. These two closest points are vertex points. And a perpendicular Bisector of the semi major axis is called a semi minor axis. The point, where these axis intersect, is the center of the hyperbola.
In Cartesian Co-ordinate system, General equation for this curve looks something like this. Here, h, k is position of the center of the hyperbola. A and b are the lengths of the semi major and semi minor axis respectively.
Now, allow point p to move in another style. This time, it keeps the ratio s p by p m fixed. And equals to one. This trajectory is a parabola. Thus, parabola is a conic, whose eccentricity is exactly one.
In Cartesian Co-ordinate system, the equation for this parabola may be like this one. Here, h, k is the co-ordinate of the vertex. And, a is the distance between the focus and the vertex. And also, vertex to the directrix.
Again, let us move the point p with a new constraint. This time it moves in such a way that the ratio of s p and p m remains fixed and less than one. In this way the trajectory it creates, is called an ellipse.
Unlike hyperbola and parabola, an ellipse is a closed curve. By symmetry, it also has two directrix and Two focal points and two vertex points. A straight line segment, connecting both vertex points, is called a major axis. This is longest diameter of an ellipse. Similarly, the shortest diameter of the ellipse is known as the minor axis.
The general equation for an ellipse is written like this. Here, h, k, is the position co-ordinate of the center of the ellipse. It is the point, where major and minor axis cross each other. Two a, and Two b are the lengths of the major and minor axis respectively.
Note that, eccentricity of an ellipse lies between zero to one.
Check that, nearer the directrix, lesser circular the curve is.
Now, move the directrix away from the focus. You can easily note that, the length of the p m is increasing. Hence eccentricity also decreasing. And the elliptic curve is becoming more and more circular. Eventually, when the directrix reaches infinity, it becomes a Circle. And its eccentricity equal to zero. The two focal point superpose each other. To become its center. Major axis and minor axis becoming equal in lengths. And introduce itself as Radius of Circle. Thus circle is a conic, whose eccentricity is zero.
General equation for a Circle is like that. Here, h, k, is the position of its center. And, r is the Radius.
Before going further. Let us summarise.
A straight line is conic with infinite eccentricity.
A hyperbola is also a conic. But its eccentricity lies between infinity and one.
If, eccentricity is exactly one, then, it is a parabola.
Ellipse is a conic, whose eccentricity lies between one and zero.
And finally, eccentricity of a circle is simply zero.
Class 11 maths. Class 11 coordinate geometry. JEE MAIN, NEET, pmt
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