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IIT JEE Coordinate Geometry- Preparation Tips

IIT JEE Coordinate Geometry
Coordinate geometry is one of the most interesting and important topics of the mathematics syllabus of JEE Advanced and JEE Main. It is one of the easiest and most scoring topic of JEE mathematics.The applications of coordinate geometry are spread through various fields of mathematics like trigonometry, calculus, dimensional geometry etc. and also have application in statistics and physics.

Important topics in coordinate geometry for IIT JEE

It is an important topic and can be further divided into important parts like:

  • 2D Geometry

Distance formula

Section formula

Area of triangle

Locus of a point

Transformation of axis

  • Straight Lines

          Concepts of straight line

  • Conic Section

Circle

Parabola

Ellipse

Hyperbola

Pair of straight lines

We shall give a brief introduction of each of these sub – topics along with the tips to master themBeginners are advised to refer the study material on coordinate geometry.

Important concepts and formulae

Distance Formula

                                                

Distance ‘d’ between any two points A (x1,y1) and B (x2,y2) on the coordinate axis is given by

                                            Distance Formula 

Section Formula                                            

Coordinates of P when it divides the line AB in the ratio m : n

                                            Section Formula           

Area of triangle

There are many formulas for area of triangle. Basically it depend on what we have given in the question.

  •  Case I: If height ‘h’ and base ‘b’ are given. Then area ‘A’ is

                                             Formula of Area of a Traingle 

  • Case II: If coordinates of points are given. Let say A(x1, y1), B(x2,y2 ) and C(x3,y3 ). Then area ‘A’ is

                                            Formula of area of a traingle

Equation of straight line

  • Case I: Point – Slope form

           Let ‘m’ be the slope of line and line is passing through point A(x1,y1). Then equation of line is

                                             Point – Slope form of Straight Line

  • Case II: Point – Point form

          Let line passing through points A(x1,y1) and B(x2,y2). Then equation of line is

                                              Point form of Straight LIne

  • Case III: Slope intercept form

          Let the slope of line be ‘m’ and intercept it made on y-axis is ‘c’. Then equation of line is

                                             Slope intercept form of Straight LIne

  • Case IV: Intercept form

          Let the line made an intercept ‘a’ and ‘b’ on x-axis and y-axis respectively. Then equation of line is

                                             Intercept form of Straight Line

  • Case V: Normal form

          Let the distance of line from the origin is ‘p’ and the angle it made with the origin is .Then equation of line is

                                             Normal form of Straight Line

Distance of a point from a line

Line:ax+by+c=0. Let the point be P(x1,y1). Then distance ‘d’ of the point P from the line is

                                             Distance of a point from a line

Distance between two parallel lines

Line 1: ax+by+c=0, Line 2: ax+by+c=0.Then distance ‘d’ between lines is

                                                Distance between two parallel lines

Concurrency of lines

Two lines are always concurrent if they are not parallel. Three lines

Line 1: a1x+b1y+c1=0 ,  Line 2: a2x+b2y+c2=0  , Line 3: a3x+b3y+c3=0 are concurrent if

                                       Concurrency of lines
Collinearity of points

Three points A(x1,y1), B(x2,y2) and C(x3,y3)are collinear if

                                              Collinearity of points

Angle between two lines

Let the slope of two lines be m1 and m2. Then angle  between two lines is

                                              Angle between two lines

Position of point w.r.t line

Position of point A(x1,y1) and B(x2,y2) w.r.t line ax+by+c=0.

If ax1+by1+c1=0 and  ax2+by2+c2=0 are of same sign, then points A and B lie on the same side of the line and if they are of opposite sign, then they lie on the opposite side of line.

Locus

Locus is a path trace by a moving or a variable point under certain given condition. If we find the equation of traced path, then the equation is the required locus.Steps to follow:

  • Assume point (h, k) whose locus is to find

  • Use the given condition

  • Remove the variable

  • After removing the variable, equation will be left in h, k and some fixed number. Finally replace ‘h’ by ‘x’ and ‘k’ by ‘y’ to get the required locus. 

CONIC SECTION

                                                       Conic Section Cutting

  • Recognition of Conics

          General form: ax2+2hxy+by2+2gx+2fy+c=0

         Recognition of Conics

  • Eccentricity

Let P be any moving point and S be the focus (fixed point) of the conic. Let PM be the perpendicular distance of the point from the directrix (fixed line) of the conic. Then eccentricity or contracity ‘e’ of the conic is defined by

          Note: ‘e’ can never be negative.

          Eccentricity in conic section

  • Circle

         Formulas of circle

  • Parabola

         Formulas of parabola

  • Ellipse

         Formulas of Ellipse

  • Hyperbola

         Formulas of Hyperbola

Some Interseting facts

  • The locus of a point from which the distance of two tangents to a circle are equal is radical axis.

  • Minimum distance between any two curves always lies along the normal.

  • PQ be the focal chord of parabola and S be the focus of the parabola. Then, SP, 2a and SQ are in Harmonic progression.

  • Tangents at the extremities of the focal chord intersect at right angle on the directrix.

  • If a triangle is formed by joining the foot of normals which are concurrent on the parabola, then centroid of parabola will always lie on the axis of parabola.

  • Locus of all those points which divide the perpendicular drawn from any point on the circle to any one of the fixed diameter in a constant ratio is an ellipse.

  • Auxiliary circle ia a circle concentric with the ellipse and having diameter equal to the major axis of the ellipse.

  • Maximum four normals can be drawn to ellipse.

  • If the two circles of different radii touchs each other externally, then the locus of centre of the third of variable radius which touches these two circles externally is hyperbola.

  • The locus of all points from where perpendicular tangents are drawn to the hyperbola is the director circle of the hyperbola.

Is coordinate geometry an important part of IIT – JEE preparation? 

Coordinate geometry is an important part of JEE maths paper. It can really improve your rank. Each year in the JEE mathematics, we have around 20% - 25% of the total marks of mathematics from this part. Among the coordinate geometry, almost 50% of the total question cover from the straight line and the circles.

Books to Refer

Tips to study Coordinate Geometry

In coordinate geometry students are less likely to commit mistake. However due to lack of the conceptual clarity and less input time to the topics, people often commit silly mistakes. This is an extremely scoring topic in JEE mathematics which can get you better rank in the JEE. Here are some tips which you must follow to score high in this branch:

  • Do not practice a single problem in coordinate geometry without drawing a figure.

  • Important results should be on your tips which might prove helpful in many of the objective problems.

  • Make a different notebook for the formulas. Write all the important formulas in your notebook on daily basis.

  • Try to memorize as much formulae as you can.

  • Read the question clearly to figure out what it actually demands. It might save your time.

  • Do not read the same concept from different books as it will lead to confusion. Always stick to one particular standard text for your concepts (Class notes or SL Loney).

  • Practice as many problems as you can from the reference books. Then try to solve the previous year JEE paper problems of coordinate geometry.

  • Try your level best to solve a problem before referring its solution.

Practical Application of Coordinate Geometry

Coordinate geometry is used extensively in daily life problems. Here are some of the examples

  • Coordinate geometry is used in astronomy particularly in computing the paths of celestial bodies like planets, comets, binary star system etc.

  • Coordinate geometry is used in the study of the paths of projectile like missile and other arms.

  • Coordinate geometry is extremely useful in the aircraft industry, specifically when dealing with the shape of an airplane's fuselage. Conic curves are used to describe the fuselage cross sections and their curvature is modified along the length of the fuselage to render a smooth yet producible surface which at the same time should allow for an efficient internal arrangement.

  • Computer programmers use coordinate geometry because most of the program they write generate PDF files. And in a PDF file the printed page is one big coordinate plane. Coordinate geometry is thus used to position elements on the page. PDF files that are produced contain text, images and line drawing, all of which are placed into position by using (x, y) coordinates, distances and slopes.

  • Coordinate geometry is also used in manipulating images. The selected image is a big plane coordinate with each colour information as a single individual point. Thus when the colours of the pictures are being manipulated, the points are changed.

  • Coordinate geometry is also used in the scanners. Scanner make use of the coordinate geometry to reproduce the exact image of the selected picture in the computer. It manipulates the points of each information in the original documents.

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