Analytical geometry and the straight line Chapter 29 FYHS-Kulai by Chtan1

Some important concepts: These are called straight line equations. FYHS-Kulai by Chtan2

3 eg. 1 A line is parallel with 2x+3y=4 and it cuts the axes at A and B. Given the area of the triangle OAB is 3 units squares, where O is the origin. Find the equation of the straight line.

FYHS-Kulai by Chtan4 eg. 2

FYHS-Kulai by Chtan5

The straight line passing through the point (h,k) and having a gradient m can be written as : FYHS-Kulai by Chtan6

The straight line AB has an equation y+2x+8=0. AB intersects the x-axis at point A and intersects the y-axis at point B. Point P lies on AB such that AP:PB = 1:3. Find (a) the coordinates of P (b) the equation of the straight line that passes through P and perpendicular to AB. FYHS-Kulai by Chtan7 eg. 3

FYHS-Kulai by Chtan8 The angle Ө between two straight lines with gradients

FYHS-Kulai by Chtan9 eg. 4

FYHS-Kulai by Chtan10 Note : the sign is chosen in order to make the perpendicular from the origin positive.

FYHS-Kulai by Chtan11 eg. 5

FYHS-Kulai by Chtan12 The equations of the bisectors of the angle between 2 straight lines :

FYHS-Kulai by Chtan13 Internal division

FYHS-Kulai by Chtan14 External division

FYHS-Kulai by Chtan15 Z C B A Z is the mid-point of BC G divides ZA in the ratio 1:2 G G is one third the way up the median from A.

FYHS-Kulai by Chtan16 How to find the distance between these 2 lines ?

FYHS-Kulai by Chtan17 Two perpendicular straight lines

Equation of a straight lines passing through the point of intersection of two given straight lines. All straight lines passing through the point of intersection of two lines : a’x+b’y+c’=0 and a”x+b”y+c”=0 are given by the equation : a’x+b’y+c’ + λ(a”x+b”y+c”)=0 where λ is any constant. FYHS-Kulai by Chtan18

FYHS-Kulai by Chtan19 Circumcircle of a triangle 外接圆 Note : intersection of the bisector of the lines is the centre of the circle.

Incircle of a triangle 内切圆 FYHS-Kulai by Chtan20 Note : intersection point of angle-of-bisector lines is the centre of the circle.

FYHS-Kulai by Chtan21 Miscellaneous Examples

PN, the perpendicular from P(3,4) to the line 2x+3y=1 is produced to Q such that NQ=PN. Find the coordinates of Q. FYHS-Kulai by Chtan22 eg. 6

Determine whether the points (3,-2), (-1,7) are on the same or opposite sides of the line 2x- 5y=13. FYHS-Kulai by Chtan23 eg. 7

ABCD is a square; A is the point (0,-2) and C the point (5,1), AC being a diagonal. Find the coordinates of B and D. FYHS-Kulai by Chtan24 eg. 8

Find the acute angle between the two lines 2y-x=3, 3y+4x=5. FYHS-Kulai by Chtan25 eg. 9

Find the distance of the point (-2,1) from the line 2y-x-7=0 FYHS-Kulai by Chtan26 eg. 10

Find the equations of the lines bisecting the angles between the lines y=3x, y=x+2. Verify that the bisectors are perpendicular. FYHS-Kulai by Chtan27 eg. 11

Find the equations of the straight lines drawn through the point (1,- 2) making angles of 45° with the x-axis. FYHS-Kulai by Chtan28 eg. 12

Find the ratio in which the line 4x-y=3 divides the line joining the points (2,-1), (-3,2). FYHS-Kulai by Chtan29 eg. 13

Prove that the lines 7x+2y=5, 6x+3y=5, 5x+4y=5 are concurrent. FYHS-Kulai by Chtan30 eg. 14

Find the equation of the straight line joining the point of intersection of the lines 4x-y=7, 2x+3y=1 to the origin. FYHS-Kulai by Chtan31 eg. 15

FYHS-Kulai by Chtan32 No need to do Ex 10b Q1, 2, 3, 4, 12 Misc Ex 10

FYHS-Kulai by Chtan33 The end