Co-ordinate Geometry

After completion of this chapter, students will be able to:

  • know about the angles between two lines, pair of straight lines and circle.
  • solve the questions of angles between two lines.
  • solve the questions of pair of straight lines.
  • solve the theorems and questions of circle.

Notes

Angle between two lines

The angle between two lines

Angle between the lines y = m1 + c1 and y = m2 + c2 and y = m2x + c2

Condition of perpendicularity

m1m2 = -1

 

Condition of Parallelism

m1 = m2

Angle between the lines A1x + B1y + C1 = 0 and A2x + B2y +C2 = 0

Condition of perpendicularity

A1A2 + B1B2 = 0

Condition of Parallelism

A1B2 = A2B1

Equation of any line parallel to ax + by + c = 0

k = -bc

Equation of any line perpendicular to ax +by +c = 0

k = ac

 

 

Pairs of straight lines

General equation of the second degree :

 ax2 + 2hxy + by2 + 2gx + 2fy + c = 0.

Circle

A circle is a closed curve such that every point on the curve is at a constant distance from a fixed point.

Centre at the origin (Standard form)

x2 + y2 = r2

Centre at any point (Central form)

(x - h)2 + (y k)2 = r2

Circle with a given diameter (Diameter form)

(x - x1) . (x - x2) +(y - y1) . (y - y2) = 0

General equation of the circle

x2 + y2 + 2gx + 2fy + c = 0

 

Equation of a circle in particular cases

When a circle touches the X - axis

(x - h)2 + (y - k)2= k2

When a circle touches the Y- axis

(x - h)2 + (y - k)2 = h2

When a circle touches both the positive axes

(x - r)2 + (y - k)2 = k2