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Co-ordinate Geometry

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Co-ordinate Geometry PDF Notes, Important Questions and Formulas

Circle Theory

BASIC DEFINITION OF CIRCLE: A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

BASIC THEOREMS AND RESULTS OF

CIRCLES

  1. Concentric circles: Circles having same centre.
  2. Congruent circles: If their radii are equal.



  3. Congruent arcs: If they have same degree measure at the centre.


CYCLIC QUADRILATERALS: A quadrilateral is called a cyclic quadrilateral if its all vertices lie on a circle.

 

 

 

Parabola

 QUESTIONS BASED ON BASIC DEFINITION & PARAMETRIC REPRESENTATION

 

CONIC SECTION

A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.

  1. The fixed point is called the FOCUS.
  2. The fixed straight line is called the DIRECTRIX.
  3. The constant ratio is called the ECCENTRICITY denoted by ‘e’.
  4. The line passing through the focus & perpendicular to the directrix is called the AXIS.
  5. A point of intersection of a conic with its axis is called a VERTEX.

 

PARABOLA

A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix).

 

TYPE OF PARABOLA

Four standard forms of the parabola are

 

Parabola

Vertex

Focus

Axis

Directrix

y2=4ax

(0,0)

(a,0)

y=0

X=-a

y2=-4ax

(0,0)

(-a,0)

y=0

X=a

x2=+4ay

(0,0)

(0,a)

x=0

Y=-a

x2=-4ay

(0,0)

(0,-a)

x=0

Y=a

(y-k)2=4a(x-h)

(h,k)

(h+a,k)

y=K

K+a-h=0

(x-p)2=4b(y-q)

(p, q)

(p,b+q)

y=P

y+b-q=0

 

 

 

Straight Line

 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 x-coordinate or the abscissa of P and y is called the y coordinate or the ordinate of the point P.

 

Special Points In a Triangle with Coordinates

 Definition: The point of concurrence of the medians of a triangle is called the centroid of the Triangle.

 

INCENTRE (I)

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

 

  1. I always lies inside the triangle.
  2. Internal angle bisector divides the base in the ratio of adjacent sides.

 

EX-CENTRES (I1, I2, I3)

Definition: The centre of the e-scribed circle which is opposite to vertices.

 

CIRCUMCENTRE (C)

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

  1. For acute angle its lies inside
  2. For obtuse angle its lies outside
  3. For right angle its lies Mid point of hypotenuse

 

ORTHOCENTRE (O)

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

  

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.

 

 

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