## List of Chapters (Std. XI Maths)

### Paper 2

1. Sets, Relations and Functions
2. Logarithms
3. Complex Numbers
4. Sequences and Series
5. Permutations and Combinations
6. Method of Induction and Binomial Theorem
7. Limits
8. Differentiation
9. Integration
10. Statistics (Measures of Dispersion)
11. Probability

## Statistics, Correlation and Regression

• ### What is Statistics?

It is a branch of mathematics. It involves collection, analysis, interpretation, presentation, and organization of data. (Dictionary Definition).

Descriptive Statistics summarizes the data with the help of few indices, such as mean (the central tendency) and standard deviation (the dispersion). Inferential Statistics draws conclusions from data that are subject to random variation. Inferential statistics uses the probability theory.

• ### Classification of Data (Numerical)

If the data values are arranged in ascending or descending order, the minimum and maximum values are revealed. The quantity represented by the numerical data is termed as a variate or a variable. Often, data values repeatedly occur in the dataset. The number of times a value occurs in the dataset is termed as the frequency. The representation using frequencies is the frequency distribution.

If the range of the data is wide, instead of mentioning individual values, they are grouped into class intervals. In general, in a class interval of ${(a-b)}$ all values ${\ge a}$ and ${ are included.

Sometimes, a cumulative frequency distribution table is prepared.

• ### Representation of Data

Histogram, Frequency Polygon and Ogive are commonly used to represent the data.

• ### Descriptive Statistics : Central Tendency

As mentioned earlier, in descriptive statistics, the central tendency and the dispersion are studied. The indices of central tendency are as follows :

Arithmetic Mean :

${\mu_x \ or \ \bar x = \frac {x_1+x_2 + \cdots + x_n}{n} =\frac 1 n \sum \limits_{i=1}^{n} x_i}$

If the data are in frequency distribution format,

${\mu_x \ or \ \bar x = \frac {\sum \limits_{i=1}^{n} x_i f_i}{\sum \limits_{i=1}^{n} f_n}}$

If the data are in grouped frequency distribution format,

${\mu_x \ or \ \bar x = A + h \times \frac {\sum \limits_{i=1}^{n} f_i u_i}{\sum \limits_{i=1}^{n} f_n},}$

where ${A}$ is the assumed mean,  generally the middle value in the dataset. ${u}$ is given by ${\frac {x-A}{h}}$. ${h}$ is the width of class interval and ${x}$ is the mid-value of the class interval.

Joint Mean of 2 distributions with ${n_1}$ and ${n_2}$ values is given by

${\frac {\mu_1 n_1 + \mu_2 n_2}{n_1 + n_2}}$

Geometric Mean :

The geometric mean is given by

${\Big( \prod \limits_{i=1}^{n} x_i^{f_i} \Big)^{1/N},}$

where ${N = \sum \limits_{i=1}^{n} f_i}$

Harmonic Mean :

Harmonic mean is the reciprocal of arithmetic mean of the reciprocals of the given values.

Median/ Positional Average :

Median is that value of variate, which divides the dataset into 2 equal parts. Thus, equal number of values exist on either side of the median. If the number of values is odd, the arithmetic mean of the two middle terms is the median.

In a cumulative frequency distribution, the median is that value of variate, whose cumulative frequency is equal to or just greater than $N/2$, where $N$ is the total number of values.

Mode :

It is the most frequently occurring value in the data-set.

• ### Descriptive Statistics : Dispersion

These indices measure the spread of the values. So, 2 datasets can have same mean, but the values may be spread over a wider range for one of them.

Mean Deviation :

It is given by

${\frac {1}{N} \sum \limits_{i=1}^{n} f_i |x_i - \mu|}$

Standard Deviation :

${\sigma = \sqrt {\frac 1 N \sum \limits_{i=1}^{n} (x- \mu)^2}}$

The square of the standard deviation is known as the variance.

Root Mean Square Deviation :

${S = \sqrt {\frac 1 N \sum \limits_{i=1}^{n} (x- A)^2},}$

where ${A}$ is any arbitrary number.

${S}$ and ${\sigma}$ are related by the following relation :

${S^2 = \sigma^2 + (\mu - A)^2}$

In grouped distributions, the standard deviation is calculated as follows :

${\sigma = h \sqrt {\frac 1 N \sum \limits_{i=1}^n f_i u_i^2 - \left (\frac {\sum \limits_{i=1}^{n} f_i u_i}{N} \right)^2}}$

Here, ${h}$ is the width of class interval, ${u = \frac {x-A}{h}}$. ${A}$ is the assumed mean, ${x}$ is the mid-value of the interval. ${N}$ is the sum of all frequencies.

• ### Moments

${r}$th moment of a distribution about the mean ${\mu}$ is denoted by ${\mu_r}$ and is given by

${\mu_r = \frac {1}{N} \sum \limits_{i=1}^{n} f_i \big (x- \mu \big )^r}$

This is similar to moment of a force about a point, where we define the moment as force ${\times}$ the perpendicular distance.

${r}$th moment of a distribution about any arbitrary number ${A}$ is denoted by ${\mu'_r}$ and is given by

${\mu'_r = \frac {1}{N} \sum \limits_{i=1}^{n} f_i \big (x- A \big )^r}$

• ### Relation between ${r}$th moment about the mean (${\mu_r}$) and ${r}$th moment about any number ${A}$, (${\mu'_r}$)

It can be shown that ${\mu_r}$ and ${\mu'_r}$ are related by the following equation :

${\mu_r = \mu'_r - ^rC_1 \mu'_{r-1} \mu'_1 + ^rC_2 \mu'_{r-2} \mu'_2 + \cdots + (-1)^r (\mu'_1)^r}$

• ### Skewness

It tells how skew the frequency distribution curve is from the symmetry.

I) Frequency curve stretches towards right : Mean to the right of mode – Right/Positively skewed

II) Frequency curve stretches towards left : Mean to the left of mode – Left/Negatively skewed

Skewness is measured by

${\frac {\mu - m}{\sigma},}$

where ${m}$ is the median.

The coefficient of skewness is given by

${\beta_1 = \frac {\mu_3^2}{\mu_2^3}}$

• ### Kurtosis

In Greek, ${kurt}$ means ${bulging}$. The coefficient of kurtosis, ${\beta_2}$ is given by

${\beta_2 = \frac {\mu_4}{\mu_2^2}}$

It measures how peaked the frequency distribution curve is.

The curve, which is neither flat nor peaked is ${mesokurtic}$. If $\beta_2 > 3$, the curve is ${leptokurtic}$ or peaked. If ${\beta_2 < 3}$, the curve is ${platykurtic}$.

• ### Bivariate Distributions

When the dataset contains 2 variables, each data point is an ordered pair, say ${(x,y)}$. Such distributions are known as bivariate distributions. We may wish to test the relationship between the 2 variables, if any.

Examples : Amount of time spent by students on social media vs. marks obtained, rainfall in a year and the crop production in the following year.

We may be tempted to come up with a conclusion without having a look at the actual values. The variables in the first example may seem to be correlated, but we cannot be very much sure unless we have the dataset.

In the second example, we can provide a good reasoning for the correlation.

• ### Correlation and Karl Pearson’s Coefficient

When a change in one variable leads to a corresponding change in the other, we say that the variables are correlated. If one increases and the other decreases, it is a negative correlation. If both increase, it is a positive correlation.

If the ratio of values of variables in each pair ${(x,y)}$ is constant, the correlation is said to be linear. According to Karl Pearson, the strength of linear relationship between the variables is given by the correlation coefficient :

${r = \frac {cov (x,y)}{\sigma_x \sigma_y} = \frac {\frac 1 n \sum \limits_{i=1}^n (x_i - \mu_x)(y_i - \mu_y)}{\sigma_x \sigma_y }}$

${cov (x,y)}$ is the covariance. When the covariance is divided by the product of the standard deviations, we get the correlation coefficient.

Note that the value of ${r}$ always lies between ${-1}$ and ${1}$.

• ### Regression

If the variables are known to be correlated, value of one variable can be obtained, if the value of the other variable is known. This is known as regression. In order to do that, a relationship between the variables, say ${x}$ and ${y}$ is developed, in the form of an equation. Further, if it is known that the relationship is linear, the equation will be of the form

${y=mx+c}$

I) Line of Regression of ${y}$ on ${x}$

${y - \mu_y = r \frac {\sigma_y}{\sigma_x} (x- \mu_x)}$

II) Line of Regression of ${x}$ on ${y}$

${x - \mu_x = r \frac {\sigma_x}{\sigma_y} (y- \mu_y)}$

### Regression Coefficients

I) of ${y}$ on ${x}$, ${b_{yx} = \frac {cov (x,y)}{\sigma_x^2}}$

I) of ${x}$ on ${y}$, ${b_{xy} = \frac {cov (x,y)}{\sigma_y^2}}$

### FOR MCQs

Almost all MCQs are formula-based. So, make sure that you know all formulas with the terms forming them.

## Set Theory

• ### Introduction

The set theory, developed by George Cantor, forms a basis for advanced mathematical topics, such as the calculus, real and complex analysis etc. It defines functions, which are an extremely important entity in mathematics.

• ### Definition

A set is a collection of well defined objects. For example, the set of all subjects studied by you in class 10. This includes English, Marathi, Hindi, French, Mathematics, Science and Social Science.

The collection of all intelligent people in your locality does not qualify as a set. The intelligence cannot be quantied and therefore, it is not a well-defined object. The set of all prime numbers less than 15 is another example. This set includes 2,3,5,7,11 and 13.

The objects which belong to a set are called elements of that set. The repetition of elements is NOT allowed in set theory. So, if there are 2 elements, which are identical, they will be treated as a single element.

• ### Representation of Set

A set is represented by capital Roman alphabets, such as A, P etc. It is customary to write down the elements of set within curly brackets. So, the set of all prime numbers less than 15 will be written as (let us call the set as P)

${P = \{ 2, 3, 5, 7, 11, 13\} }$

This representation of a set, where we list down all the elements, is known as the Roster form. The same set can also be written in Set-builder form as follows:

${P = \{ x \ | \ x \ is \ a \ prime \ number \ less \ than \ 15 \} }$

• ### Sets of Numbers

The set of natural numbers, also known as counting numbers, is denoted by

${\mathbb {N} = \{ 1, 2, 3, \cdots \} }$

The set of whole numbers is denoted by

${\mathbb {W} = \{0, 1, 2, 3, \cdots \} }$

Thus, ${\mathbb {W}}$ is made up of the number zero as well as ${ \mathbb {N}}$.

The set of integers is denoted by

${\mathbb {Z} = \{\cdots, -3, -2, -1, 0, 1, 2, 3, \cdots \}}$

The next set of numbers is the set of rational numbers. These numbers can be represented in the form ${\frac p q}$ , where ${p}$ and ${q}$ are any integers and ${q \ne 0}$. It is denoted by ${\mathbb {Q}}$.

There are numbers, such as ${\sqrt 2, \sqrt 3, e}$, which cannot be represented in the form ${\frac p q}$ . Such numbers are known as irrational numbers. The set of irrational numbers is denoted by ${\mathbb I}$.

When the set of rational numbers ${\mathbb {Q}}$ and the set of irrational numbers ${\mathbb I}$ are combined together, the set of real numbers, ${\mathbb R}$ is obtained.

• ### Terminology

When all the elements of set ${A}$ and set ${B}$ are identical, we say that ${A}$ and set ${B}$ are equal. So,

${A = B}$

In roster form, equal sets are identical; but their set-builder forms may differ. Note that they are one and the same.

An empty set (or a null set) is the one, which contains no element. It is denoted by

${\{ \ \} \ or \ \phi}$

For example, ${\{x | x \in \mathbb N \ and \ 4.

A set with only 1 element in it is known as singleton set. For example, ${\{x | x \in \mathbb N \ and \ 4. This contains only 1 element, which is ${5}$.

Set ${A}$ is said to be a subset of set ${B}$, when all elements in set ${A}$ are contained in set ${B}$. Set ${B}$ is then called the superset of ${A}$. We denote this by,

${A \subseteq B \ and \ B \supseteq A}$

For example, ${A = \{1, 2, 5, 6 \}, \ B = \{-3,0,1,1.5,2,4,5,6,8\} }$. Notice that ${B}$ contains all elements which are there in ${A}$.

A non-empty set ${P}$ is said to be a proper subset of a set ${Q}$, such that there exists at least one element in ${Q}$, which does not belong to ${P}$. i.e.

${P \subset Q, \ when \ \exists \ x,\ x \in Q, \ x \ne P}$

A set, which has a finite number of elements is called a finite set. e.g.

${P = \{ x | x \in \mathbb N, \ x < 6}$

A set, where the process of counting elements does not end is known as an infinite set. e.g. the set of natural numbers, ${\mathbb N}$.

A universal set is the one, which in a discussion, serves as a parent set. In other words, all other sets are subsets of this set. Note that, for all kinds of numbers we discussed earlier, the set of real numbers, ${\mathbb R}$ is the universal set. Universal set is generally denoted by ${U}$.

• ## Operations on sets

• Union

Union of 2 sets ${A}$ and ${B}$ is the set of all elements, which are present either in ${A}$ or in ${B}$ or in both ${A}$ and ${B}$. It is denoted by

${A \cup B}$

• Intersection

Intersection of 2 sets ${A}$ and ${B}$ is the set of all elements, which are common to both A and B. It is denoted by

${A \cap B}$

• Complement of a set

Let ${A}$ be any set. The complement of set ${A}$ is the set of all elements, which do not belong to ${A}$; but are contained in the universal set ${U}$. It is denoted by ${A^c}$ or ${A^0}$.So,

${A \cap A^c = \phi , \ A \cup A^c = U}$

• ### Power set

The power set of a set ${A}$ is the set of ALL subsets of ${A}$, including the null set and ${A}$ itself. So, if ${A = \{ 2,4,5 \}}$, the power set will be

${P = \{ \ \{ \ \}, \{2 \}, \{4 \}, \{5 \}, \{2,4 \}, \{2,5 \}, \{ 4,5 \}, \{ 2,4,5 \} \ \}}$

Note that if a set has ${n}$ elements, its power set will have ${2^n}$ elements. So, power set of A will have ${2^3=8}$ elements.

• ### Venn Diagrams

A Venn diagram of a set is the pictorial representation of the set. Generally, Venn diagrams are geometrical figures, such as circle, triangle, ellipse etc. The operations on set can be represented readily using these diagrams.

• ### De Morgan’s laws

${(A \cup B)' = A' \cap B'}$

${(A \cap B)' = A' \cup B'}$

• ### Few More Terms

• Commutativity of Operations:

${A \cup B = B \cup A }$

${A \cap B = B \cap A}$

• Associativity of Operations:

${A \cup (B \cup C )= (A \cup B) \cup C}$

${A \cap (B \cap C )= (A \cap B) \cap C}$

• Idempotent laws

${A \cap A = A, A \cup A = A}$

• Distributive Properties of Operations

${A \cap (B \cup C) = (A \cap B) \cup (A \cap C)}$

${A \cup (B \cap C) = (A \cup B) \cap (A \cup C)}$

## Sequences and Series

We use the words sequence and series interchangeably and we generally mean a continuation of numbers/events by them. In mathematics, however, these words have a particular meaning.

A sequence (also known as progression) is an arrangement of numbers in a specific order, in such a way that a definite relation exists between the numbers and their positions.

A series is what one gets, when all the terms of a sequence are added.

The numbers which make a sequence are known as the terms. The ${n}$th term is generally denoted by ${t_n}$.

Summing up the terms of the sequence ${t_1, t_2, t_3, t_4, t_5, \cdots , t_n}$, we get a series ${S_n}$.

It is easy to prove that ${t_n = S_n - S_{n-1}}$.

• ### WHY do we study sequences and series?

1) The investment on compound interest grows annually forming a geometric sequence.

2) Population models of bacteria (2 from 1, 4 from 2 etc.) follow a geometric sequence.

3) Difficulty level of problems increases in such a way that each problem takes 5 minutes more than the previous one to be solved. This follows an arithmetic sequence.

4) The harmonic sequence ${1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5 + \cdots}$ diverges to ${\infty}$, and is a fundamental result in mathematics. It has got applications in estimating traffic and in destroying noise!

Terms of a sequence form a function defined from ${\mathbb N}$, the set of natural numbers to ${t_n}$, the set containing the terms in the sequence; e.g. consider the sequence ${1,4,9,16,25, \cdots}$. Clearly, ${t_n = f(n) = n^2 \ \forall n \in \mathbb N}$.

• ### Arithmetic Progression (A.P.)

Arithmetic Progression is a sequence for which ${t_{n+1}-t_n}$ is constant ${\forall n \in \mathbb N}$. The constant difference , ${d}$, is known as the common difference and generally, the first term is denoted by ${a}$.

So, ${n}$th term will be ${a+ (n-1)}d$ and sum of first ${n}$ terms will be

${\frac {n}{2} [2a + (n-1)d]}$

• ### Properties of A.P.

Let ${k}$ be a non-zero number. Let ${a, a+d, a+2d, \cdots}$ be an A.P. Then,

I) ${a+k, a+d + k, a+2d + k, \cdots}$ is also an A.P. with first term ${a+k}$ and common difference ${d}$

II) ${ak, (a+d)k, (a+2d)k, \cdots}$ is also an A.P. with first term ${ak}$ and common difference ${dk}$.

• ### Geometric Progression (G.P.)

Geometric Progression is a sequence for which ${\frac {t_{n+1}}{t_n}}$ is constant ${\forall n \in \mathbb N}$. The constant ratio , ${r}$, is known as the common ratio.

If ${a}$ is the first term of a G.P., then the ${n}$th term is ${ar^{n-1}}$ and the sum of first ${n}$ terms is given by

${a \Big ( \frac {r^n-1}{r-1} \Big) , r \ne 1}$

• ### Properties of G.P.

Let ${k}$ be a non-zero number. Let ${a, ar, ar^2, \cdots}7s=1$ be a G.P. Then,

I) ${ka, kar, kar^2, kar^3, \cdots}$ is also a G.P. with first term ${ka}$ and common ratio ${r}$

II) ${a^k, (ar)^k, (ar^2)^k, (ar^3)^k, \cdots}$ is also a G.P. with first term ${a^k}$ and common ratio ${r^k}$.

Having studied limits, we can show that if ${|r| < 1}$, then sum of all terms of a G.P. (up to infinity) is given by

${\frac {a}{1-r}}$

In other words,

${\lim \limits_{n \to \infty} S_n = \frac {a}{1-r}}$

• ### Harmonic Progression (H.P.)

Harmonic Progression is a sequence such that reciprocals of its terms are in arithmetic progression. Thus, if ${t_1, t_2,t_3, \cdots, t_n, \cdots}$ is a harmonic progression, then

${\frac {1}{t_1}, \frac {1}{t_2}, \frac {1}{t_3}, \cdots, \frac {1}{t_n} , \cdots}$

is an A.P.

Arithmetic mean, ${A}$, of ${x}$ and ${y}$ is given by ${\frac {x+y}{2}}$. It can be shown that

${x, \frac {x+y}2, y}$

are in A.P.

Geometric mean,${G}$, of ${x}$ and ${y}$ is given by ${\sqrt {xy}}$. It can be shown that

${x, \sqrt {xy}, y}$

are in G.P.

The harmonic mean, ${H}$, of ${x}$ and ${y}$ is given by ${\frac {2xy}{x+y}}$. Thus,

${\frac 1 x , \frac {x+y}{2xy}, \frac 1 y}$

are in A.P.

It can be shown that

${G^2 = AH \ and \ A \ge G \ge H}$

An arithmetico-geometric progression is a combination of an A.P. and a G.P. It is of the form

${a, (a+d)r, (a+2d)r^2, (a+3d)r^3 , (a+4d)r^4}$

Sum of first ${n}$ terms of such a progression is given by

${S_n = \frac {a}{1-r} + \frac {dr(1-r^{n-1})}{(1-r)^2} - \frac {r^n [a+(n-1)d]}{1-r}}$

• ### The summation notation ${\sum}$

Let there be a sequence ${t_1, t_2, t_3, t_4, \cdots, ... t_m}$. The sum of these $m$ terms is briefly written as

${\sum \limits_{r=1}^m t_r = t_1 + t_2 + t_3 + t_4 + \cdots + t_m}$

• Properties of the summation notation

${\sum \limits_{r=1}^m at_r = a \sum \limits_{r=1}^m t_r}$

${\sum \limits_{r=1}^m a_r + b_r \sum \limits_{r=1}^m a_r + \sum \limits_{r=1}^m b_r}$

• Important results :

I) Sum of first ${n}$ natural numbers is given by

${\frac {n(n+1)}{2}}$

II) Sum of squares of first ${n}$ natural numbers is given by

${\frac {n(n+1)(2n+1)}{6}}$

III) Sum of cubes of first ${n}$ natural numbers is given by

${\frac {n^2(n+1)^2}{4}}$

• ### Exponential and logarithmic series

I) The exponential series ${e^x}$ is an infinite series given by

${\sum \limits_{r=0}^{\infty} \frac {x^r}{r!} = 1 + \frac {x}{1!} + \frac {x^2}{2!} + \frac {x^3}{3!} + \cdots + \frac {x^n}{n!} + \cdots }$

II) For ${|x| < 1}$,

${ln (1+ x) = \sum \limits_{r=1}^{\infty} (-1)^{r+1} \frac {x^r}{r} = x - \frac {x^2}{2} + \frac {x^3}{3} - \frac {x^4}{4} + \cdots}$

## Hyperbola

NOTE: We are going to use the section formula to derive the equation of hyperbola, as we did for ellipse. Since there are 2 variants of the formula (external and internal division), we get 2 foci and 2 directrices.

A hyperbola is a conic section, whose eccentricity ${e}$ is greater than 1. In other words, if ${S(ae,0)}$ is a focus and ${x= \frac {a}{e}}$ is the directrix, then for any point ${P(x,y)}$,

${SP = e PM, \ and \ e>1}$

The standard equation of hyperbola is

${\frac {x^2}{a^2}- \frac {y^2}{b^2}=1}$

• Hyperbola is NOT a closed curve, has 2 parts, which are mirror images of each other
• Symmetric about ${X}$ and ${Y}$ axes
• Does not pass through the origin
• Intersection with ${X}$ axis – at ${(a,0)}$ and ${(-a,0)}$
• Does not intersect ${Y}$ axis
• Foci : ${(ae,0)}$ and ${(-ae,0)}$, directrices : ${x \pm \frac a e =0}$
• Difference of focal distances = ${2a}$ = constant (We’ve used this as the condition for locus of a point)
• Length of latus rectum = ${\frac {2b^2}{a}}$
• Parametric Equations : ${x = a sec (\theta), \ y= b tan (\theta)}$
• Transverse axis has length ${2a}$conjugate axis has length ${2b}$
• If ${a=b}$, we get a rectangular hyperbola

• ### Where can you see the hyperbolic shape?

I) The black lines on a basketball or the red lines on the baseball

II) Orbits of comets around the Sun (or any star)

III) Interference patterns by 2 circular waves

IV) Potato chips (:P)

V) Cooling towers in an industrial plant

## Ellipse

NOTE: We are going to use the section formula to derive the equation of ellipse. Since there are 2 variants of the formula (external and internal division), we get 2 foci and 2 directrices.

An ellipse is a conic section, whose eccentricity ${e}$ is less than 1. In other words, if ${S(ae,0)}$ is a focus and ${x= \frac {a}{e}}$ is the directrix, then for any point ${P(x,y)}$,

${SP = e PM, \ and \ e<1}$

The standard equation of ellipse is

${\frac {x^2}{a^2}+ \frac {y^2}{b^2}=1}$

• Ellipse is horizontal, when ${a>b}$ i.e. more stretched in ${X}$ direction, vertical, when ${b>a}$
• Closed curve
• Symmetric about ${X}$ and ${Y}$ axes
• Does not pass through the origin
• Intersection with ${X}$ axis – at ${(a,0)}$ and ${(-a,0)}$
• Intersection with ${Y}$ axis – at ${(0,b)}$ and ${(0,-b)}$
• Foci : ${(ae,0)}$ and ${(-ae,0)}$, directrices : ${x \pm \frac a e =0}$
• Sum of focal distances = ${2a}$ = constant (We’ve used this as the condition for locus of a point)
• Length of latus rectum = ${\frac {2b^2}{a}}$
• Parametric Equations : ${x = a cos (\theta), \ y= b sin (\theta)}$

• ### Where can you find the elliptic shape?

I) Earth’s orbit around Sun, with Sun as one of the foci of the ellipse

II) The rugby ball

III) Batman logo (boundary)

IV) Eggs, lemons

V) Whispering galleries

## Parabola

The word parabola originated as a combination of 2 Greek words, para, which means besides and bole, which means throw. When any heavy object is thrown in air, its trajectory is a parabola. Hence the name.

The eccentricity of parabola, ${e}$ is ${1}$. Let ${S(a,0)}$ be the focus and ${d \equiv x+a=0}$ be the directrix. According to the focus-directrix property,

${SP = e PM \ and \ e =1, \ \therefore \ SP= PM,}$

where ${P(x,y)}$ is any point on the parabola. Using this condition for locus of a point, we get the standard equation of parabola,

${y^2 = 4ax, \ a>0}$

• It is symmetric about ${X}$ axis, and it extends to infinity to the right of ${Y}$ axis
• Focal distance = ${x_1 + a}$
• Latus rectum = ${4a}$
• Parametric equations : ${x= at^2, \ y=2at}$, where ${t}$ is the parameter
• The general equation of parabola is of the form ${y=ax^2+bx+c}$ or ${x=ay^2+by+c , a \ne 0}$. It can be converted to the standard form by shift of origin and (sometimes) rotation of axes.

• ### Where can you see the parabolic shape?

I) Shape of satellite dish

II) Automobile Headlights (The dim-dip feature)

III) McDonald’s Arches

IV) Mirror Furnace (capable of producing temperatures up to ${3300^oC}$ from rays of the Sun)