Understanding Engineering Mathematics

7.1.5 Parallel and perpendicular lines ➤214 222➤➤

A.Find the equation of the straight line parallel to the liney= 1 − 3 xand which passes
through the point(− 1 , 2 )
B.Find the equation of the straight line perpendicular to the line through the points
(− 1 , 2 ), (0, 4) and passing through the first point.

7.1.6 Intersecting lines ➤216 222➤➤

Find all points where the following straight lines intersect.

(i) x+y= 3 (ii) 2x+ 2 y=− 1 (iii) y= 3 x− 1

7.1.7 Equation of a circle ➤217 222➤➤

A.(a) Write down the equation of the circles with the centres and radii given

(i) (0, 0), 2 (ii) (1, 2), 1 (iii) (−1, 4), 3

(b) Determine the centre and radius of the circles given by the equations:

(i) x^2 +y^2 − 2 x+ 6 y+ 6 = 0 (ii) x^2 +y^2 + 4 x+ 4 y− 1 = 0

7.1.8 Parametric representation of curves ➤219 222➤➤

Eliminate the parametertand obtain the Cartesian equation of the curve, in terms ofx

andy:

(i) x= 3 t+1,y=t+ 2 (ii) x=6cos2t,y=6sin2t

(iii) x=2cost−1,y=2sint (iv) x=cos 2t,y=cost

7.2 Revision

7.2.1 Coordinate systems in a plane

➤

204 220➤

In Chapters 5 and 6 we never really had to fix our geometric objects in the plane in any

way. A point such as the centre of a circle is special so far as the circle is concerned,

but where it is located in the plane has no bearing on the properties of the circle. This

characterises what is calledsyntheticoraxiomaticgeometry, in which the description of

geometric objects such as lines and circles depends on their relation to each other.

However, by locating points in space by means of acoordinate system, assigning

numbers, orcoordinatesto each point, we can represent geometrical ideas bymathematical

equations. This is the subject ofcoordinate geometry– also calledanalytical geometry.

The simplest and most common coordinate system is theCartesian coordinate system

orrectangular coordinate systemin a plane. Anoriginis chosen and two perpendicular

lines through the origin are chosen as thex-andy-axes(91

➤

). The scale on each axis

is taken to be the same in this chapter. Any point in the plane can then be specified by

giving itsxcoordinate andycoordinate as an ordered pair(x, y). See, for examples, the

solution to the review question.