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JEE Advanced 2023 Revision Notes for Analytical Geometry

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JEE Advanced 2023 Revision Notes for Analytical Geometry - Free PDF Download

One of the scoring chapters in JEE Advanced Maths syllabus is analytical geometry. This chapter introduces the concepts of straight lines, triangles, centroid, circumcentre, the intersection of straight lines, orthocentre, etc in coordinate geometry. It will also teach how to determine the angle between two intersecting lines too. It means this chapter will explain the concepts related to straight lines in coordinate geometry and other structures related to straight lines. To make the preparation easier, refer to the JEE Advanced Mathematics Analytical Geometry notes prepared by the experts.

Category:

JEE Advanced Revision Notes

Content-Type:

Text, Images, Videos and PDF

Exam:

JEE Advanced

Chapter Name:

Analytical Geometry

Academic Session:

2024

Medium:

English Medium

Subject:

Mathematics

Available Material:

Chapter-wise Revision Notes with PDF

The experts of Vedantu have formulated an easier version of all the mathematical principles of coordinate geometry related to straight lines, triangles, angles, etc. These notes will enable you to understand the concepts better and will help you prepare them faster. Download and add these notes to your analytical geometry study material for study preparation today.

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Access JEE Advanced Revision Notes Mathematics Analytic Geometry

Coordinate System:

The three mutually perpendicular lines in a space divide the space into eight parts, and if these perpendicular lines are the coordinate axes, then it is said to be a coordinate system.

If we want to do more than draw on a sheet of paper or in abstract three dimensional space we will need a way to assign more meaning to what the X, Y and possibly Z numbers are supposed to mean in a particular case, for example, how they might be used to specify a location on the Earth's surface.
                         
                        

Coordinate System


Distance Between Two points:

Let $P\left( {{x_1},{y_1},{z_1}} \right)$  and $Q\left( {{x_2},{y_2},{z_2}} \right)$  be two given points. The distance between these points is given by

$PQ\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}}$ 

The distance of a point P(x, y, z) from origin O is

$OP = \sqrt {{x^2} + {y^2} + {z^2}}$


Section Formulae:

  • The coordinates of any point, which divides the join of points $P\left( {{x_1},{y_1},{z_1}} \right)$  and $Q\left( {{x_2},{y_2},{z_2}} \right)$  in the ratio $m:n$  internally are $\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$ 

  • The coordinates of any point, which divides the join of points $P\left( {{x_1},{y_1},{z_1}} \right)$ and $Q\left( {{x_2},{y_2},{z_2}} \right)$  in the ratio m : n externally are $\left( {\dfrac{{m{x_2} - n{x_1}}}{{m - n}},\dfrac{{m{y_2} - n{y_1}}}{{m - n}},\dfrac{{m{z_2} - n{z_1}}}{{m - n}}} \right)$

  • The coordinates of mid-point of P and Q are $\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2},\dfrac{{{z_1} + {z_2}}}{2}} \right)$

  • Coordinates of the centroid of a triangle formed with vertices $P\left( {{x_1},{y_1},{z_1}} \right)$  and $Q\left( {{x_2},{y_2},{z_2}} \right)$ and $R\left( {{x_3},{y_3},{z_3}} \right)$  are $\left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_2} + {y_3}}}{3},\dfrac{{{z_1} + {z_2} + {z_2}}}{3}} \right)$

The Centroid of a Tetrahedron:

If $\left( {{x_1},{y_1},{z_1}} \right)$ , $\left( {{x_2},{y_2},{z_2}} \right)$ , $\left( {{x_3},{y_3},{z_3}} \right)$  and $\left( {{x_4},{y_4},{z_4}} \right)$  are the vertices of a tetrahedron, then its centroid G is given by $\left( {\dfrac{{{x_1} + {x_2} + {x_3} + {x_4}}}{4},\dfrac{{{y_1} + {y_2} + {y_3} + {y_4}}}{4},\dfrac{{{z_1} + {z_2} + {z_3} + {z_4}}}{4}} \right)$


Direction Cosines:

If a directed line segment OP makes angle α, β and γ with OX , OY and OZ respectively, then $\cos \alpha$ , $\cos \beta$  and $\cos \gamma$  are called direction cosines of up and it is represented by l, m, n. i.e.,

 $l = \cos \alpha$ 

$m = \cos \beta$ 

$n = \cos \gamma$

C:\Users\Untitled.21\Downloads\drive-download-20220622T092321Z-001\Direction Cosines.png


Direction Cosines

If OP = r, then coordinates of OP are (lr, mr, nr)

If l, m, n are direction cosines of a vector r, then 

$\begin{array}{l}r = |r|\left( {li + mj + nk} \right)\\ \Rightarrow r = li + mj + nk\\{l^2} + {m^2} + {n^2} = 1\end{array}$ 

Projections of r on the coordinate axes are $|r| = l|r|,m|r|,n|r|$ Sum of the squares of projections of r on the coordinates axes if $P\left( {{x_1},{y_1},{z_1}} \right)$ and $Q\left( {{x_2},{y_2},{z_2}} \right)$ are two points, such that the directions cosines of PQ are l, m, n then

${x_2} - {x_1} = 1|PQ|,~{y_2} - {y_1} = m|PQ|,~{z_1} - {z_2} = n|PQ|$ 

These are projections of PQ on X,Y and Z axes, respectively.

If l, m, n are directions cosines of a vector r and a, b, c are three numbers, such that $\dfrac{l}{a} = \dfrac{m}{b} = \dfrac{n}{c}$ .

Then, we say that the direction ratio of r are proportional to a,b,c.

Also, we have

$l = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$ ,$m = \dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$ ,$n = \dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$ 


If $\theta$  is the angle between two lines having direction cosines ${l_1},{m_1},{n_1}$ and ${l_2},{m_2},{n_2}$ Then $\operatorname{\cos} \theta=l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}$

Lines are parallel, if $\dfrac{{{l_1}}}{{{l_2}}} = \dfrac{{{m_1}}}{{{m_2}}} = \dfrac{{{n_1}}}{{{n_2}}}$ 

Lines are perpendicular, if ${l_1}{l_2} + {m_1}{m_2} + {n_1}{n_2}$ 

If $\theta$  is the angle between two lines whose direction ratios are proportional to ${a_1},{b_1},{c_1}$ and${a_2},{b_2},{c_2}$  then the angle $\theta$  between them is given by

$\cos \theta  = \dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}$ 

Lines are parallel, if $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$ 

Lines are perpendicular, if ${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0$ 

The projection of the line segment joining points $P\left( {{x_1},{y_1},{z_1}} \right)$  and $Q\left( {{x_2},{y_2},{z_2}} \right)$ to the line having direction cosines l, m, n is $|\left( {{x_2} - {x_1}} \right)l + \left( {{y_2} - {y_1}} \right)m + \left( {{z_2} - {z_1}} \right)n|$ 

The direction ratio of the line passing through points $P\left( {{x_1},{y_1},{z_1}} \right)$  and $Q\left( {{x_2},{y_2},{z_2}} \right)$ are proportional to ${x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}$. Then, direction cosines of PQ are $\dfrac{{{x_2} - {x_1}}}{{|PQ|}},\dfrac{{{y_2} - {y_1}}}{{|PQ|}},\dfrac{{{z_1} - {z_1}}}{{|PQ|}}$ 

There are different types of coordinates in analytic geometry. Some of them are as follows:

  • Cartesian Coordinates

  • Polar Coordinates

  • Cylindrical Coordinates 

  • Spherical Coordinates

Cartesian Coordinates:

The most well-known coordinate system is the Cartesian coordinate, where every point has an x-coordinate and y-coordinate expressing its horizontal and vertical positions, respectively. They are usually addressed as an ordered pair and denoted as (x, y). We can also use this system for three-dimensional geometry, where every point is represented by an ordered triple of coordinates (x, y, z) in Euclidean space.


Polar Coordinates:

In the case of polar coordinates, each point in a plane is denoted by the distance ‘r’ from the origin and the angle θ from the polar axis.


Cylindrical Coordinates:

In the case of cylindrical coordinates, all the points are represented by their height, radius from the z-axis and the angle projected on the xy-plane with respect to the horizontal axis. The height, radius and angle are denoted by h, r and θ, respectively.


Spherical Coordinates:

In spherical coordinates, the point in space is denoted by its distance from the origin ( ρ), the angle projected on the xy-plane with respect to the horizontal axis (θ), and another angle with respect to the z-axis (φ).


Cartesian Plane:

In coordinate geometry, every point is said to be located only on the coordinate plane or Cartesian plane.

Coordinate Plane in Analytic Geometry


Cartesian Plane


The above graph has an x-axis and y-axis as its Scale. It is similar to the box explained above. The x-axis runs across the plane, and the Y-axis runs at the right angle to the x-axis.


Origin:

 It is the point of intersection of the axis(x-axis and y-axis). Both x and y-axis are zero at this point.


Values of the Different Sides of the Axis:

x-axis : 

The values on the right-hand side of this axis are positive, and those on the left-hand side are negative

y-axis:

The values above the origin are positive and below the origin are negative.

To Locate a Point:

We need two numbers to locate a plane in order of writing the location of the X-axis first and the Y-axis next. Both will tell the single and unique position on the plane. You must compulsorily follow the order of the points on the plane. (x, y)

If you look at the figure above, point A has a value of 3 on the x-axis and 2 on the Y-axis. These are the rectangular coordinates of Point A represented as (3, 2).


Distance Formula:

Let the two points A and B having coordinates to be$\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ .

Thus the distance between two points are

$d = \sqrt {\left[ {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \right]}$


Midpoint theorem Formula:

Let A and B are some points in a plane, which is joined to form a line, having     coordinates (x1, y1) and (x2, y2), respectively. Suppose, M(x, y) is the midpoint of the line connecting the point A and B then its formula is given by;

$M(x,y) = \left[ {\dfrac{{\left( {{x_1} + {x_2}} \right)}}{2},\dfrac{{\left( {{y_1} + {y_2}} \right)}}{2}} \right]$


Angle Formula:

Let two lines have slope m1 and m2 and θ is the angle formed between the two lines A and B, which is represented as:

$\tan \theta  = \dfrac{{\left( {{m_1} - {m_2}} \right)}}{{\left( {1 + {m_1}{m_2}} \right)}}$


Section Formula:

Let two lines A and B have coordinates (x1, y1) and (x2, y2), respectively. A point P the two lines in the ratio of m:n, then the coordinates of P is given by;

When the ratio m:n is internal:

$\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + m{y_1}}}{{m + n}}} \right)$ 

When the ratio m:n is external:

$\left( {\dfrac{{m{x_2} - n{x_1}}}{{m - n}},\dfrac{{m{y_2} - m{y_1}}}{{m - n}}} \right)$


Angle Between Two Intersecting Lines:

If $l(x_1, m_1, n_1) ~\text{and}~ l(x_2, m_2, n_2)$  be the direction cosines of two given lines, then the angle θ between them is given by

$\cos \theta  = {l_1}{l_2} + {m_1}{m_2} + {n_1}{n_2}$ 

The angle between any two diagonals of a cube is ${\cos ^{ - 1}}\left( {\dfrac{1}{3}} \right)$ 

The angle between a diagonal of a cube and the diagonal of a face.


Straight Line in Space:

  • The two equations of the line $ax + by + cz + d = 0$ and $a'x + b'y + c'z + d' = 0$ together

  • Represents a straight line.

  • Equation of a straight line passing through a fixed point $A\left( {{x_1},{y_1},{z_1}} \right)$ and having through ratios a, b, c is given by

  • $\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c}$ ,it is also called the symmetrically from of a line.

  • Any point P on this line may be taken as $\left( {{x_1} + \lambda a,{y_1} + \lambda b,{z_1} + \lambda c} \right)$ , where λ ∈ R is parameter. If a, b, c are replaced by direction cosines 1, m, n, then λ, represents distance of the point P from the fixed point A.

  • Equation of straight line joining two fixed points $A\left( {{x_1},{y_1},{z_1}} \right)$ and $B\left( {{x_2},{y_2},{z_2}} \right)$ is given by

$\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_2}}}$ 

  • Vector equation of a line passing through a point with position vector a and parallel to vector b is $r = a + \lambda b$ where A is a parameter.

  • Vector equation of a line passing through two given points having position vectors a  and b is$r = a + \lambda b(b - a)$ where λ is a parameter.

  • The length of the perpendicular from a point $P\left( {\overrightarrow \alpha  } \right)$ on the line $r - a + \lambda b$ is given by

$\sqrt {|\overrightarrow \alpha   - a| - {{\left\{ {\dfrac{{\left( {\overrightarrow \alpha   - a} \right).b}}{{|a|}}} \right\}}^2}}$ 

  • The length of the perpendicular from a point $P\left( {{x_1},{y_1},{z_1}} \right)$ on the line

$\dfrac{{x - a}}{l} = \dfrac{{y - b}}{m} = \dfrac{{z - c}}{n}$ is given by

  • $\sqrt {\left\{ {{{\left( {a - {x_1}} \right)}^2} + {{\left( {b - {y_1}} \right)}^2} + {{\left( {c - {z_1}} \right)}^2}} \right\} - {{\left\{ {\left( {a - {x_1}} \right)l + \left( {b - {y_1}} \right)m + \left( {c - {z_1}} \right)n} \right\}}^2}}$ 

  • where, l, m, n are direction cosines of the line.

Skew Lines:

Two straight lines in space are said to be skew lines, if they are neither parallel nor intersecting.

  • The shortest distance between lines$r = {a_1} + \lambda {b_1}$ and $r = {a_2} + \mu {b_1}$ is given by

              $d = |\dfrac{{\left( {{b_1} \times {b_2}} \right).\left( {{a_2} - {a_1}} \right)}}{{|{b_1} \times {b_2}|}}|$ 

  • The shortest distance parallel lines$r = {a_1} + \lambda {b_1}$ and $r = {a_2} + \mu {b_1}$ is given by

$d = |\dfrac{{\left( {{a_2} - {a_1}} \right) \times b}}{{|b|}}|$ 

  • Lines$r = {a_1} + \lambda {b_1}$ and $r = {a_2} + \mu {b_1}$ are intersecting lines, if $\left( {{b_1} \times {b_2}} \right) \times \left( {{a_2} - {a_1}} \right) = 0$ 

  • The image or reflection $\left( {x,y,z} \right)$  of a point $\left( {{x_1},{y_1},{z_1}} \right)$  in a plane $ax + by + cz + d = 0$  is given by

$\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c} = \dfrac{{ - 2\left( {a{x_1} + b{y_1} + c{z_1} + d} \right)}}{{{a^2} + {b^2} + {c^2}}}$ 

  • The foot  $\left( {x,y,z} \right)$ of a point $\left( {{x_1},{y_1},{z_1}} \right)$  in a plane$ax + by + cz + d = 0$  is given by

$\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c} = \dfrac{{ - \left( {a{x_1} + b{y_1} + c{z_1} + d} \right)}}{{{a^2} + {b^2} + {c^2}}}$ 

  • Since, x, y and z-axes pass through the origin and have direction cosines (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively. Therefore, their equations are

           x-axis: $\dfrac{{x - 0}}{1} = \dfrac{{y - 0}}{0} = \dfrac{{z - 0}}{0}$ 

          y-axis: $\dfrac{{x - 0}}{0} = \dfrac{{y - 0}}{1} = \dfrac{{z - 0}}{0}$ 

          z-axis: $\dfrac{{x - 0}}{0} = \dfrac{{y - 0}}{0} = \dfrac{{z - 0}}{1}$ 

Normal Form of the Equation of Plane:

  • The equation of a plane at a distance p from the origin and the direction cosines of the normal from the origin to the plane are l, m, n is given by lx + my + nz = p.

  • The coordinates of the foot of perpendicular N from the origin on the plane are (lp, mp, np).

Intercept Form:

The intercept form of equation of plane represented in the form of

$\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1$ 

where, a, b and c are intercepts on X, Y and Z-axes, respectively.

Example: The distance of the line $3y - 2z - 1 = 0 = 3x - z + 4$ from the point (2,-1,6) is

Solution:

$3y - 2z - 1 = 0 = 3x - z + 4$

$3y - 2z - 1 = 0$

Directions $\Rightarrow \left( {3, - 1,0} \right)$

Let’s direction of given line are a, b, c

Now, 3b - 2c = 0 and 3a - c = 0

6a = 3b = 2c

Since, a : b : c = 3 : 6 : 9

Any point on line be 3K-1, 6K+1, 9K+1

Now, 3(3K-1) + 6(6K+1)1 + 9(9K+1) = 0

$K = \dfrac{1}{3}$

Point on line $\Rightarrow$ (0, 3, 4)

Given point ( 2, -1, 6)

Distance $= \sqrt {4 + 16 + 4}$

$= 2\sqrt 6$


Example: Let the actual angle bisector of the two planes $x - 2y - 2z = 0$ $2x - 3y - 6z + 1 = 0$be the plane P. Then which points lie on P?

Solution:

${P_1} = x - 2y - 2z + 1 = 0$

${P_2} = 2x - 3y - 6z + 1 = 0$

$\left| {\dfrac{{x - 2y - 2z + 1}}{{\sqrt {1 + 4 + 4} }}} \right| = \left| {\dfrac{{2x - 3y - 6x + 1}}{{\sqrt {{2^2} + {3^2} + {6^2}} }}} \right|$

$\dfrac{{x - 2y - 2z + 1}}{3} =  \pm \dfrac{{2x - 3y - 6z + 1}}{7}$

Since, ${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 20 > 0$

The negative sign will give an acute bisector

$7x - 14y - 14z + 7 =  - \left[ {6x - 9y - 18z + 3} \right]$

$ \Rightarrow 13x - 23y - 32z + 10 = 0$

$\left( { - 2,0, - \dfrac{1}{2}} \right)$


Importance of Maths Analytical Geometry

Coordinate geometry is very crucial for the preparation for the JEE Advanced competitive exam. This chapter teaches students about the concepts of straight lines, and their equations to represent different types of straight lines, interceptions, angles between straight lines, centroids, circumcentre, orthocentre, etc.

The concepts will be explained in the form of mathematical expressions using the old fundamental principles students studied in the previous classes. They will learn how to prove concurrency and other features of straight lines using those formulae.

These formulae will be used to solve problems related to coordinate geometry at an advanced level. This chapter is also very important for engineering aspirants. Apart from answering the fundamental questions of JEE Advanced exams, these concepts will be used lifelong in engineering streams.

This chapter covers the various geometric aspects of a circle, conic section, 3D geometry, and other principles of coordinate geometry. The theories and applications will be explained systematically. To understand the importance of this chapter, use the Analytical Geometry JEE Advanced notes.


Benefits of Analytical Geometry JEE Advanced Notes

  • You can refer to the easier explanation of all the advanced concepts of this chapter given by the subject matter experts. Find out the easiest ways to prepare this chapter by comprehending the derived formulae from the notes.

  • Download for free and use these notes at your convenience. You can easily make your preparation sessions more fruitful with the help of these concise notes of analytical geometry. Take a step ahead and use these notes to revise this chapter before the exam.

  • You can now easily recall what you have studied from the organised presentation of this chapter in these notes. The summarised version of the formulae and mathematical principles in these notes will give you a better preparation platform for the IIT JEE competitive exam.

  • Practice the sample questions given in these notes to check your preparation level. Evaluate your answers by comparing them to the solutions. Learn the best techniques to find accurate answers in no time.

Download Free Analytical Geometry JEE Advanced Notes PDF

Why wait then? You can download the free PDF version of these revision notes and access them at your convenience. Make your preparation better and faster. Learn how to utilise your time to complete preparing analytical geometry. Remember what you have studied by simply checking Analytical Geometry JEE Advanced revision notes without facing any difficulty. Solve all doubts on your own and gain more confidence to score well in the JEE Advanced exam.


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FAQs on JEE Advanced 2023 Revision Notes for Analytical Geometry

1. What is a straight line?

The shortest distance between two points is called a straight line.

2. What is an orthocentre?

It is defined as the intersection point of the three altitudes of a triangle drawn from the vertices to the opposite side. It means that an orthocentre is the intersection point of three altitudes. If it is an acute-angled triangle, the orthocentre will remain inside the triangle. An obtuse-angled triangle will have its orthocentre outside the triangle.

3. What is the distance between two parallel lines?

The length of a perpendicular drawn from any point of a straight line touching the other one is the distance between these two parallel straight lines.

4. What is the slope of a triangle?

The angle made by a straight line with the positive X-axis in coordinate geometry is called the slope of the straight line.