eSaral provides detailed Notes of Physics, Chemistry, Mathematics and Biology Notes for class 11, and 12. So here you will get class 11 notes for mathematics. There are important points in Mathematics such as formulae, equations, identities, properties, theorem etc. what has to be remembered to solve problems in Math.eSaral is providing complete study material to prepare for IIT JEE, NEET and Boards Examination. So here are Parabola Class 11 Notes & Numericals for IIT JEE Exam preparation. With the help of Notes, candidates can plan their Strategy for a particular weaker section of the subject and study hard. So, go ahead and check the Important Notes for CBSE Class 11 Maths.
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The vertex of a parabola is the point at the intersection of the parabola and the line of symmetry. It is the point where the parabola makes its sharpest turn. It will denote the maximum and minimum points of the parabola. For a parabola y2 = 4ax, the vertex is (0, 0).
Candidates who are pursuing in CBSE Class 11 Maths are advised to revise the notes from this post. With the help of Notes, candidates can plan their Strategy for particular weaker section of the subject and study hard. So, go ahead and check the Important Notes for CBSE Class 11 Maths Parabola from this article.
Let P be a point lying within or outside a given parabola. Suppose any straight line drawn through P intersects the parabola at Q and R. Then, the locus of the point of intersection of the tangents to the parabola at Q and R is called the polar of the given point P with respect to the parabola and the point P is called the pole of the polar.
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Remember: the vertex is the parabola's maximum or minimum point. We often refer to its coordinates with the letters \(h\) and \(k\) and say that it has coordinates: \(\begin{pmatrix}h,k\end{pmatrix}\).
If a>0 in f( x )=a x 2 +bx+c, the parabola opens upward. In this case the vertex is the minimum, or lowest point, of the parabola. A large positive value of a makes a narrow parabola; a positive value of a which is close to 0 makes the parabola wide.
A parabola is a graph of a quadratic function. Pascal stated that a parabola is a projection of a circle. Galileo explained that projectiles falling under the effect of uniform gravity follow a path called a parabolic path. Many physical motions of bodies follow a curvilinear path which is in the shape of a parabola.
In mathematics, any plane curve which is mirror-symmetrical and usually of approximately U shape is called a parabola. Here we shall aim at understanding the derivation of the standard formula of a parabola, the different equations of a parabola, and the properties of a parabola.
A parabola refers to an equation of a curve, such that each point on the curve is equidistant from a fixed point, and a fixed line. The fixed point is called the "focus" of the parabola, and the fixed line is called the "directrix" of the parabola. Also, an important point to note is that the fixed point does not lie on the fixed line. Thus, a parabola is mathematically defined as follows:
The four standard forms are based on the axis and the orientation of the parabola. The transverse axis and the conjugate axis of each of these parabolas are different. The following are the observations made from the standard form of equations:
Let us consider a point P with coordinates (x, y) on the parabola. As per the definition of a parabola, the distance of this point from the focus F is equal to the distance of this point P from the Directrix. Here we consider a point B on the directrix, and the perpendicular distance PB is taken for calculations.
Normal: The line drawn perpendicular to tangent and passing through the point of contact and the focus of the parabola is called the normal. For a parabola y2 = 4ax, the equation of the normal passing through the point \((x_1, y_1)\) and having a slope of m = -y1/2a, the equation of the normal is \((y - y_1) = \dfrac{-y_1}{2a}(x - x_1)\)
Chord of Contact: The chord drawn to joining the point of contact of the tangents drawn from an external point to the parabola is called the chord of contact. For a point \((x_1, y_1)\) outside the parabola, the equation of the chord of contact is \(yy_1 = 2x(x + x_1)\).
Pole and Polar: For a point lying outside the parabola, the locus of the points of intersection of the tangents, draw at the ends of the chords, drawn from this point is called the polar. And this referred point is called the pole. For a pole having the coordinates \((x_1, y_1)\), for a parabola y2 =4ax, the equation of the polar is \(yy_1 = 2x(x + x_1)\).
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Parabola is an important curve of the conic section. It is the locus of a point that is equidistant from a fixed point, called the focus, and the fixed line is called the directrix. Many of the motions in the physical world follow a parabolic path. Hence learning the properties and applications of a parabola is the foundation for physicists.
The standard equation of a parabola is y2 = 4ax. The axis of this parabola is the x-axis which is also the transverse axis of the parabola. The focus of the parabola is F(a, 0), and the equation of the directrix of this parabola is x = -a.
The equation of the parabola can be derived from the basic definition of the parabola. A parabola is the locus of a point that is equidistant from a fixed point called the focus (F), and the fixed-line is called the Directrix (x + a = 0). Let us consider a point P(x, y) on the parabola, and using the formula PF = PM, we can find the equation of the parabola. Here the point 'M' is the foot of the perpendicular from the point P, on the directrix. Hence, the derived standard form of parabola is y2 = 4ax.
The eccentricity of a parabola is equal to 1 (e = 1). The eccentricity of a parabola is the ratio of the distance of the point from the focus to the distance of this point from the directrix of the parabola.
The line perpendicular to the transverse axis of the parabola and passing through the vertex of the parabola is called the conjugate axis of the parabola. For a parabola y2 = 4ax, the conjugate axis is the y-axis.
The standard equation of a parabola is used to represent a parabola algebraically in the coordinate plane. The general equation of a parabola can be given as, y = a(x-h)2 + k or x = a(y-k)2 +h, where (h,k) denotes the vertex. The standard form of parabola is y2 = 4ax or x2 = 4ay.
All parabolas are not necessarily a function. Parabolas that open upwards or downwards are considered parabolic functions. The parabolas that open to left or right side fail vertical line test and hence are NOT functions.
A conic curve has a wide range of applications in the real world. Conic curves can be seen in buildings, temples, mosques, footballs, ice-cream cones, etc. There is a specific chapter on Conic sections in class 11 which elucidates upon the applications of conic curves in various areas such as planetary motion, design of telescopes and antennas, reflectors in flashlights and automobile headlights, etc. If you are studying this chapter and need help in understanding the key pointers, here are the notes and summary on Conic sections.
To identify which kind of equation is used in the conic section in class 11 maths question paper, you can use the following formula. But begin by changing the equation into this form- Ax2 +Bxy+Cy2+Dx+Ey+F=0
A collection of all the points that lie at an equal distance with respect to a stationary point in a particular plane is simply referred to as a circle. Moreover, it is also an ellipse having both the foci on a single point that is its centre. Here are the parts of the circle as mentioned in class 11 conic sections chapter:
As per the class 11 conic sections, we define a parabola as an accumulation of all points in a plane that lies at an equal distance from a stationary line and a stationary point (that is not on the line). The stationary line is called the directrix of the parabola, and the stationary point F is called the focus. The axis of a parabola is the line going through the focus and at a right angle to the directrix.
Ellipse is defined as an oval-shaped figure. We can explain ellipse as a closed conic section having two foci (plural of focus), made by a point moving in such a manner that the addition of the length from two static points (two foci) does not vary at any point of time. Here are some essential points related to an ellipse that are derived from the class 11 conic sections chapter:
Class 11 maths NCERT book states that a hyperbola is a combination of all points in a plane such that the difference between distances to two static points (foci) in a plane is constant at every point. As per the conic sections chapter of class 11, hyperbola has two stationary points that are known as the focus of the hyperbola. Have a look at the important pointer related to it- 17dc91bb1f
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