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Notes (Solutions) of Unit 06: Conic Section, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.

When we observe the meaning of Hyperbola, it seems precisely like two parabola's opposite to each other. But that is not the case despite having the same curve structure; these two are very different from each other. In Hyperbola, we calculate how distant a set of points are from two fixed points whereas in the parabola a set of points is at equal distances from the directrix. One major difference between them is that all parabolas have the same shape whereas all hyperbolas have different shapes. You may also notice that the two arms in a parabola are parallel, but that is not the case for hyperbolas.

Here is the complete notes of Hyperbola Class 11Math JEE Main and Advanced, NEET and school exams for the chapter for you to read online. Bookmark this page and read it online whenever you want. Alternatively you can download it in PDF format from above to read it offline whenever you want.

Let $$S_1$$ denote the set of all pairs (x, y) of real numbers that fulfill the condition $$x^2 - y^2 = 1$$, and $$S_2$$ denote the set of all pairs (x, y) of real numbers that fulfill the condition $$x^2 + y^2 = 1 \,$$. In this paper we consider quadratic real functions f that satisfy the additional equation $$y^2 f(x) = x^2 f(y)$$ under the condition $$(x,y) \in S_i$$$$(i=1,2)$$. We prove that each of these conditions implies $$f(x) = f(1) x^2$$ for all $$x \in \mathbb R$$.

Free download NCERT Solutions for Class 11 Maths Chapter 11 Conic Sections Ex 11.1, Ex 11.2, Ex 11.3, Ex 11.4 and Miscellaneous Exercise PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

Ex 11.4 Class 11 Maths Question 8.Vertices (0, 5), foci (0, 8)Solution:Vertices are (0, 5) which lie on x-axis. So the equation of hyperbola in standard form

Ex 11.4 Class 11 Maths Question 9.Vertices (0, 3), foci (0, 5)Solution:Vertices are (0, 3) which lie on x-axis. So the equation of hyperbola in standard form

The optimum stacking hyperbola described by equation (15) is not necessarily the small-spread hyperbola given by equation (14). Refer to the travel-times illustrated in Figure 3.1-16 and note the following:

In practice, when we refer to stacking velocity and the zero-offset time associated with the optimum stacking hyperbola described by equation (15), we almost always think of the moveout velocity and the zero-offset time associated with the small-spread hyperbola given by equation (14).

NCERT solutions for Class 11 maths Chapter 11 exercise 11.4 Conic Sections explains the curve hyperbola, which is a very interesting and equally useful geometric shape. It has got two curves, two focal points, two vertices, and two axes, so it's a property-rich shape. Any point on the hyperbola shares a common property with all the other points in the hyperbola, that is the difference of the distance from the two focal points is constant. This commonality helps us to get the equation of the hyperbola x2/a2 - y2/b2 = 1; for this hyperbola, the center is in the origin, i.e., the coordinates are (0, 0) with (-a, 0) and (a, 0) as the two vertices.

There are 15 questions in this NCERT solutions Class 11 maths Chapter 11 exercise 11.4 that help kids in determining the equations of a hyperbola and identify the various components of it like the foci, vertices, eccentricity, and length of the latus rectum. In a few questions, the coordinates of vertices and foci are given in order to find the equation of the hyperbola.

There are certain variations wherein instead of giving the coordinates of the vertices, the students will be provided with the length of the transverse axis, conjugate axis, or the length of the latus rectum along with the coordinates of the foci and henceforth will be asked to find the equation of the hyperbola. The pdf block of the exercise questions in the Class 11 maths NCERT solutions Chapter 11 exercise 11.4 Conic Sections is given below :

NCERT solutions Class 11 maths Chapter 11 exercise 11.4 has questions that will make the students understand what hyperbola is, what are its components, and the relation of its components with each other. Kids should focus on understanding the equation of a hyperbola in depth and how to apply it to a range of sums.

NCERT solutions Class 11 maths Chapter 11 exercise 11.4 ensures that the students are comfortable with the shape of the hyperbola and appreciate the unique relationship shared by all points in the hyperbola. The questions in this exercise are a mere play of having one clue or another and finding out the missing answer.

We study the distribution of points on the $(n+1)$-dimensional modular hyperbola $a_1\cdots a_n+1 \equiv c \pmod q$, where $q$ and $c$ are relatively prime integers. In particular, we show that an elementary argument leads to a straight-forward proof of a recent result of T.Zhang and W.Zhang, with a stronger error term. We also use character sums to obtain an asymptotic formula for the number of points in a given box that lie on such hyperbolas.

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