COORDINATE GEOMETRY – EQUATIONS OF LINES
ASSIGNMENT 3: EQUATION OF A LINE FROM TWO POINTS
Question 1:
A straight line, PQR, joins the points P(-1, -1), Q(3, 7) and R(x, y), where
Q is the midpoint of the line PR. Another line, L, is drawn at point R and
passes through the point S(1, -4). Find the equation of the line L.
First we need to find the coordinates of R. If Q is the midpoint then R must be
(7, 15) by adding another 4 to the x coordinate and 8 to the y coordinate.
We then use R(7, 15) and S(1, -4) to find the equation of the required line.
y – 15/-4 -15 = x – 7/ 1 -7
y – 15/-19 = x – 7 / -6
Cross multiply to get:
-6(y - 15) = -19(x – 7)
-6y + 90 = -19x + 133
Take everything to one side to get:
19x – 6y – 43 = 0
Question 2:
The line that passes through the points (2, -5) and (-7, 4) meets the xaxis at the point P. Work out the coordinates of P.
Submitted by Cathy-Anne Murphy:
(2,-5) = (x1,y1)
(-7,4) = (x2,y2)
y--5/4--5 = x-2/-7-2
-9(y+5) = 9(x-2)
-9y + 45 = 9x - 18
0 = 9x + 9y -18 +45
0 = 9x + 9y + 27
0 = x + y + 3
---------------------The last part of the question was to find where this line crosses the x-axis,
therefore we let y=0 in the equation to get:
x+0+3=0
x = -3
Therefore coordinates of P are (-3, 0).
Question 3:
The line y = 2x - 10 meets the x-axis at the point A. The line y = -2x +
4 meets the y-axis at the point B. Find the equation of the line joining A
and B.
We first need to find the coordinates of A and B.
For A: it meets the x-axis so we let y = 0 to get:
2x – 10 = 0
2x = 10
x=5
Coordinates are (5, 0)
For B: the line meets the y-axis so we let x = 0 to get:
y=4
Coordinates are (0, 4)
Substitute coordinates into formula to get:
y – 0/ 4 – 0 = x – 5/ 0 – 5
y / 4 = x – 5/ -5
Cross multiply to get:
-5y = 4x – 20
Take everything to one side and let it equal to 0:
4x + 5y – 20 = 0
Question 4:
The lines y = -2x + 1 and y = x + 7 intersect at the point L. The point M
has coordinates (-3, 1). Find the equation of the line joining L and M.
Submitted by Miceal McCann:
y= -2x + 1
y= x + 7
Solving equations simultaneously to find coordinates of L.
x+7 = -2x+1
3x = -6
x = -2
y=x+7
y = -2 + 7 = 5
Coordinates are (-2, 5)
(-3,1)
(-2,5)
y-1= x--3
5-1=-2--3
y-1= x+3
4=1
y-1= 4x+12 => 4x-y+13=0
Question 5:
The line V passes through the points (-5, 3) and (7, -3) and the line W
passes through the points (2, -4) and (4, 2). The lines V and W intersect
at the point A. Work out the coordinates of the point A.
Submitted by Johnny Feenan:
Eq. for V
m= diff. in y over diff. in x
= (-3-3) / (7--5)
= -6/12 = -1/2
so y= -1/2X+c
sub co-ord. in (7,-3)
-3= -1/2(7) + c
-3= -3.5+c
c= -3+ -3.5=1/2
V is: y= -1/2x+1/2
Eq for W
m= diff. in y over diff. in x
=(2--4) / (4-2)
=6/2= 3
so y=3x+c
sub. in co-ord. (4,2)
2=3(4)+c
c=2-12= -10
W is: y=3x-10
rearrange both eq.
V: 1/2x+y=1/2
W: 3x-y=10
solve simultaneously by adding:
3x – y = 10
1/2x+ y = 1/2
3.5x = 10.5
x = 10.5/3.5 = 3
if x is equal to 3
3(3) – y = 10
y = 9 – 10 = -1
so point A: (3,-1)
----------------Alternative Method:
Use the formula for two points – (y – y1)/(y2 – y1) = (x – x1)/(x2 – x1) to get
equations for V and W and then solve simultaneously.
Question 6:
(a) Find the equation of the line L which passes through the points A(1, 0)
and B(5, 6).
(b) The line M with equation 2x + 3y = 5 meets l at the point C. Determine
the coordinates of C.
Submitted by Bernadette Byrne, Jamie Cunningham and Michael Boyle:
(a) y – y1 = x – x1
A(1,0) B(5,6)
y2 – y1 x2 – x1
y – 0= x – 1
6–0 5–1
y–0=x–1
6
4
4(y – 0) = 6(x – 1)
4y – 0 = 6x – 6
4y – 6x + 6 = 0
(b) Need to solve the two equations simultaneously:
-6x + 4y + 6 = 0
2x + 3y – 5 = 0
(1)
(2)
Multiply (2) by 3 and then add the two equations together:
-6x + 4y + 6 = 0
6x + 9y – 15 = 0
13y – 9 = 0
13y = 9
y = 9/13
To find corresponding x value we substitute y = 9/13 into either equation to get:
2x + 3(9/13) – 5 = 0
2x = 38/13
x = 19/13
Coordinates of C are (19/13, 9/13).