Coordinate Geometry JEE Notes | EduRev

Mathematics (Maths) Class 11

Created by: Tarun Kaushik

JEE : Coordinate Geometry JEE Notes | EduRev

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A. COORDINATE GEOMETRY

Coordinate Geometry is the unification of algebra and geometry in which algebra is used in the study of geometrical relations and geometrical figures are represented by means of equations. The most popular coordinate system is the rectangular Cartesian system. Coordinates of a point are the real variables associated in an order to describe its location in space. Here we consider the space to be two-dimensional.

Through a point O, referred to as the origin, we take two mutually perpendicular lines XOX' and YOY' and call them x and y axes respectively. The position of a point is completely determined with reference to these axes by means of an ordered pair of real numbers (x, y) called the coordinates of P where | x | and | y | are the distances of the point P from the y-axis and the x-axis respectively, x is called the xcoordinate or the abscissa of P and y is called the y-coordinate or the ordinate of the point P.

Coordinate Geometry JEE Notes | EduRev

(1) Distance between two points :

(a) Let A and B be two given points, whose coordinates are given by A(x1, y1) and B(x2 , y2) respectively. 
Coordinate Geometry JEE Notes | EduRev

(b) Distance of (x1, y1) from origin :  Coordinate Geometry JEE Notes | EduRev

Note :- If two vertex A(x1, y1), B(x2, y2) are given then third vertex of equilateral triangle

Coordinate Geometry JEE Notes | EduRev

(2) Section formula :

Coordinates of the point P dividing the join of two points A(x1, y1) and B(x2, y2) internally in the given  ratio λ1 : λ2

Coordinate Geometry JEE Notes | EduRev

Coordinates of the point P dividing the join of two points A(x1, y1) and B(x2, y2) externally in the ratio λ1 : λ2 

Coordinate Geometry JEE Notes | EduRev

(3) Special points in a triangle with co-ordinates :

(a) Centroid (G) 

Definition : The point of concurrence of the medi ans of a triangle is called the centroid of the triangle.

Coordinate Geometry JEE Notes | EduRev

(i) G divides median into 2 : 1.

(ii) G always lies inside the triangle.

(iii) Co-ordinates of G is  Coordinate Geometry JEE Notes | EduRev

(b) Incentre (I) :

Definition : The point of concurrency of the internal bisectors of the angles of a triangle is called the incentre of the triangle. 

Coordinate Geometry JEE Notes | EduRev

(i) I always lies inside the triangle.

(ii) Internal angle bisector divides the base in the ratio of adjacent sides.

(iii) Co-ordinates of I is  

Coordinate Geometry JEE Notes | EduRev

where a, b, c are the lengths of the sides of the Δ

(c) Ex-centres (I1, I2, I3) :

Definition : The centre of the escribed circle which is opposite to vertices.

To get I1 (or I2 or I3) replace a by –a (b by –b or c by –c) in formula of coordinate of I

(d) Circumcentre (C) :

Definition : The point of concurrency of the perpendicular bisectors of the sides of a triangle is called circumcentre of the triangle.

Coordinate Geometry JEE Notes | EduRev

(i)    For acute angle Δ ⇒  lies inside

(ii)   For obtuse angle Δ ⇒  lies outside

(iii)  For right angle Δ ⇒  Mid point of hypotenuse

(iv)  Co-ordinates of circumcentre is

Coordinate Geometry JEE Notes | EduRev

(e) Orthocentre (O) :

Definition : The point of concurrency of the altitudes of a triangle is called the orthocentre of the triangle.

Coordinate Geometry JEE Notes | EduRev

Coordinate Geometry JEE Notes | EduRev

Coordinate Geometry JEE Notes | EduRev

Notes :

(i) In any triangle O, G, C are collinear.

(ii) In any triangle G divides the line joining O & C in ratio 2 : 1.

(iii) In an equilateral triangle O, G, C, I are coincident.

(iv) In an isosceles triangle O, G, C, I are collinear.

(f) Harmonic Conjugate : If P is a point that divides AB internally in the ratio m1: m2 and Q is another point which divides AB externally in the same ratio m1 : m2, then the point P and Q are said to be Harmonic conjugate to each other with respect to A and B.

Coordinate Geometry JEE Notes | EduRev

Coordinate Geometry JEE Notes | EduRev

Note :- Internal and External angle bisector of an angle divides the base harmonically.

Ex. If midpoints of the sides of a triangle are (0, 4), (6, 4) and (6, 0), then find the vertices of triangle, centroid and circumcentre of triangle.

Sol. Let points A(x1, y1), B(x2, y2) and C(x3, y3) be vertices of ΔABC.

Coordinate Geometry JEE Notes | EduRev

Coordinate Geometry JEE Notes | EduRev

Solving we get A(0, 0), B (12, 0) and C(0, 8).

Hence ΔABC is right angled triangle ∠A = π/2

Circumcentre is midpoint of hypotenuse which is (6, 4) itself and

Coordinate Geometry JEE Notes | EduRev

Ex.2 Prove that the incentre of the triangle whose vertices are given by A(x1, y1), B(x2, y2), C(x3, y3) is Coordinate Geometry JEE Notes | EduRev where a, b, and c are the sides opposite to the angles  A, B and C respectively.

Sol.

By geometry, we know that BD/DC = AB/AC 

If the length of the sides AB, BC and AC are c, a and b respectively, then  BD/DC = AB/AC  = c/b

Coordinate Geometry JEE Notes | EduRev

Coordinate Geometry JEE Notes | EduRev

Coordinate Geometry JEE Notes | EduRev

Coordinate Geometry JEE Notes | EduRev

 

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