JEE Advanced 2023 Revision Notes for Analytical 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.

                         

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.,

 $l = \cos \alpha$ 

$m = \cos \beta$ 

$n = \cos \gamma$

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.

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

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