Chapter 6 Coordinate Geometry Module Exercises

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CHAPTER 6: COORDINATE GEOMETRY 6.1 DISTANCE BETWEEN POINTS Distance between points A( x1 , y1 ) and B( x 2 , y 2 ) is given by the formula:

(x2 − x1 )2 + ( y2 − y1 )2 Exercise 1 1.

Find the distances between the points: (a)

(3, 2) and (8, 14)

(b)

(0, 4) and (–9, 1)

[Ans: (a) 13 units (b) 9.487 units]

2.

The distance between the points K (19, a ) and L (4,3) is 17 units. Find the possible values of a. [Ans: –5, 11]

3.

Given the distance between (p, 2p) and (1, 6) is 10 units. Find the possible values of p. [Ans: −

9

,7 ]

5

4.

Given the points R (k ,3), S (−2,1) and T (3,2) . Find the possible values of k if the length RS is twice the length of ST. [Ans: –12, 8]

5.

If the point G ( p, q) is equidistant from the points A(2,−1) and B (3,6) , show that p = 20 − 7 q.

6.

The vertices of a triangle are H (0,3), J (4,6) and L(−3,7) . Calculate the perimeter of the triangle HJL. [Ans: 17.07 units]

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6.2 (a) MIDPOINT BETWEEN TWO POINTS The midpoint between points A( x1 , y1 ) and B ( x 2 , y 2 ) is given by the formula:

x +x y +y  2, 1 2  1  2 2  

Exercise 2 7.

Write down the coordinates of the midpoints of the straight lines joining the points: (a)

A(5,−2) and B (−3, 6)

(b)

1 1 1  1 R ,−1  and T  5 ,−4  2 2 2  2

(c)

V ap 2 ,2ap and W aq 2 ,2aq

(

)

(

[Ans: (a) (1, 2) (b) (3,–3) (c)

8.

)

 ap 2 + aq 2   , ap + aq  ]   2  

D, E and F are three points on a straight line with E(3, 5) as the midpoint of DF. Given that D is (− 2, 3) , find the coordinates of F. [Ans: (8, 7)]

9.

The points P and Q are (α ,2) and (− 2, β ) respectively. The midpoint of PQ is R (1, 1) . Determine the value of α and the value of β . [Ans: α = 4, β = 0 ]

10.

P(3, 1) , Q(m,−1) , R(0, 2) and S (4, n ) are the vertices of a parallelogram PQRS, calculate the values of m and of n. [Ans: m = −1, n = 4 ]

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6.2 (b) DIVISION OF A LINE SEGMENT The coordinates of a point that divides a line segment according to a given ratio m : n is given by the fomula:

m A( x1 , y1 )

n C ( x, y )

B( x2 , y 2 )

Coordinates of

+ mx2 ny1 + my2  , m+n m + n 

 nx

C =  

1

Exercise 3 11.

Find the coordinates of a point which divides the straight line joining the following pairs in the given ratio. (a)

(− 1,−2) and (4, 2)

(b)

(− 3, 1) and  − 1 ,−3   3

ratio = 4 : 1 

ratio = 3 : 5

-

1  1  [Ans: (a)  3, 1  (b)  − 2, ] 2  5 

12.

Find the coordinates of the point P(x, y) which divides the line joining A(2, 1) and B(7, 9) such that AP : PB = 2 : 3. 1  [Ans:  4, 4  ] 5 

13.

P and Q are the points (6, 4) and (16, 9) respectively. W is a point on the straight line PQ such that 3PW=2WQ. Find the coordinates of W. [Ans: (10, 6)]

14.

The point G lies on the straight line KL where K and L are (4, 3) and (8, 7) respectively. If KG : GL = 6 : t – 1, find the coordinates of G in terms of t.  4t + 44 , 3t + 39  ] [Ans:    t +5 t+5 

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15.

1  F  5, 3  divides the line joining A(2, 8) and B(6, 2) internally in the ratio m : n. 2  Find the value of m and of n. [Ans: m = 3, n = 1]

16.

The point T ( f , 5) lies on the line joining Q (1, 2) and V (−4, 9) . Find (a)

QT , TV

(b)

the value of f.

[Ans: (a)

3 4

, (b) −

8

]

7

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6.3 AREA OF POLYGONS y

B( x 2 , y 2 )

A( x1 , y1 )

x

O

C ( x3 , y 3 ) The area of triangle ABC can be calculated by the formula: Area ∆ABC =

1 x1 2 y1

x2

x3

x1

y2

y3

y1

= ( x1 y 2 + x 2 y 3 + x3 y1 ) − ( x 2 y1 + x3 y 2 + x1 y 3 ) = _________ units2

Exercise 4 17.

Find the area of the given triangles with its vertices. (a)

K (−4, 6), R(2, 1), T (3, 5)

(b)

H (−5,−2), J (3,−3), L(1, 4)

[Ans: (a) 14.5 units2, (b) 27 units2]

18.

The vertices of a triangle ABC are (0, 0), (6, 0) and (p, q) respectively. Given the area of ∆ABC = 15 units 2 , find the possible values of p and q. [Ans: p = 5, q = –5, p = –5, q = 5]

19.

Find the area of the following quadrilaterals. (a)

E (2, 7), F (7, 8), G (9, 3), H (0, 2)

(b)

W (1, 4), X (−3, 1), Y (0,−4), Z (5,−1)

[Ans: (a) 35 units2, (b) 33 units2]

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20.

Show that the following points (–3, 6), (1, –2) and (2, –4) lies on a straight line(collinear).

21.

The points (4, m), (0, 2), (–2, –3) lie on a straight line, find the value of m. [Ans: m = 12]

22.

The area of the quadrilateral KLMN where K (2t , 0), L(1, 3), M (4, − t ) and N (0,−3) is 23 units2. Find the value of t given that t is negative. [Ans: t = –2]

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6.4 EQUATION OF A STRAIGHT LINE Learning Outcomes




Determine the x-intercept and y-intercept of a line.




Find the gradient of a line passes through two points.

y 2 − y1 x 2 − x1 Find the gradient of a straight line using the x-intercept and y-intercept. y − intercept m=− x − intercept Find the equation of a straight line given: the gradient and one point, using the formula y = mx + c , two points, using the formula y = mx + c or y − y1 = m( x − x1 ) , x-intercept and y-intercept. gradient, m =











Find the gradient and intercepts of a straight line given the equation. Change the equation of a straight line to the general form, ax + by + c = 0 .




Find the point of intersection of two lines – through simultaneous equations.

Exercise 5 23.

State the values of the x-intercept and y-intercept from the following diagrams. (a)

y

(b)

y O

3

(5,0)

x

( 0, −2 )

O

x

2

y

(c)

y

(d)

( 0,3) (−6,0)

(−3,0)

O

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O ( 0, −2 )

x

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x

8

24.

25.

Find the gradient of a straight line that passes through two points as follows: (a)

(−1, 2) and (2, 6)

(b)

(4,−7) and (−1,−2)

(c)

(−5, 0) and (−4,−3)

Find the gradient of the straight line in each of the following diagrams. y

(a)

y

(b)

( 4,5)

4

x

3

O

y

(c)

y

(d) ( 4, 4 )

(1, 2 )

2

O

27.

x

x

O

26.

x

O 1

( 4, −3)

Find the equation of a straight line that has (a)

a gradient 3 and passes through the point (−4,2) ,

(b)

a gradient

(c)

a gradient −

2 and passes through the point (1,−3) , 3 1 and passes through the point (−2,−6) . 4

Find the gradient of a straight line given below: (a)

x y − = −2 3 2

(b)

3x y + =1 2 4

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28.

29.

Find the equation of a straight line joining two points as follows: (a)

(−4,3) and (2,−3) ,

(b)

(−2,−5) and (−1,6) ,

(c)

(−4,−3) and (0,−4) .

Find the equation of the straight line in each of the following diagrams. y

(a)

y

(b)

5

4

O −1 3

O

x

y

(c)

(d)

O

y −2

x

O

3

−4

x

x −5

30.

State the x-intercept and y-intercept for each of the following straight lines. (a)

y = 5x − 3

(b)

2 y = 3x + 8

(c)

2 x + y = −7

(d)

3x − 2 y = 6

(e)

2 x + 5 y − 10 = 0

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31.

Express each of the following equations in intercept form,

x y + = 1 . Hence, state a b

the values of the x-intercept, y-intercept and its gradient.

32.

33.

34.

35.

(a)

y = 1 − 5x

(b)

2y = x − 2

(c)

2x − y = 6

(d)

4 x − 7 y + 21 = 0

(e)

3 x + 2 y = 12

Express each of the following equations in general form. (a)

3 y = 5x + 1

(b)

x y = +1 2 3

(c)

4 y − 2x = −

(d)

3x +

1 3

y =2 −2

Find the point of intersection of two lines with equations as follows: (a)

y = 2 x + 4 and y − x = 5 ,

(b)

x + 2 y = 1 and

(c)

2 x − y − 6 = 0 and 3 x + 2 y + 5 = 0 .

x − 4 = 3y , 2

2  The straight line MR intersects the x-axis at  ,0  and the y-axis at (0,−1) . The 3  straight line NS has gradient 2 and passes through (1,−1) . Calculate the coordinates of the point of intersection of MR and NS. [Ans: (4, 5)]

Show that the point of intersection of the lines 3 x − 5 y + 22 = 0 2 x + 7 y − 6 = 0 also lies on the line 4 y − x − 12 = 0 .

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6.5 PARALLEL AND PERPENDICULAR LINES Learning Outcomes




Determine whether two straight lines are parallel when the gradients of both lines are known and vice versa.




Find the equation of a straight line that passes through a fixed point and is parallel to a given line.




Determine whether two straight lines are perpendicular when the gradients to both lines are known and vice versa.




Determine the equation of a straight line that passes through a fixed point and is perpendicular to a given line.




Solve problem involving equations of straight lines.

L1 Short Notes

•

Lines L1 and L2 are parallel, so m L1 = m L2 (same gradient).

•

Lines L3 and L4 are perpendicular to each other (there intersect each other at 90 o ), so, m L3 × m L4 = −1

L4

90

o

L2

L3

Exercise 6

36.

37.

Determine whether the two straight lines in each of the following are parallel. (a)

Two straight lines that pass through the points K ( 2, 4) and P ( −1, 6) and the points H (1,−3) and Q (−5, 1) .

(b)

Two straight lines that pass through the points R (−2, 5) and S (1,−1) and the points D (1, 4) and E (3, 1) .

(c)

y = 2 x − 1 and 2 y = 4 x + 2 .

(d)

x y + = 1 and 2 y = −3 x − 5 2 3

Determine whether the straight line 2 x − 3 y + 7 = 0 and the straight line 6 x − 9 y − 1 = 0 are parallel.

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38.

39.

Given that the following two straight lines are parallel, find the value of unknown, k. k x −5. 2

(a)

y = 4 x + 3, y =

(b)

x y + = 0, 3 y − kx − 4 = 0 2 4

Given the points O (0,0), A(t , t 2 ), B(t + 1, t ) and C (2t , t + 2) , find the values of t if OA is parallel to BC. [Ans: t = −2, 1]

40.

41.

Find the equation of each of the following straight lines that passes through (a)

P(−4, 3) and is parallel to the straight line y = 2 x + 9 ,

(b)

Q ( −1,−3) and is parallel to the straight line

(c)

A(−2,3) and is parallel to the straight line joining points (5, 8 ) and (−3,2) .

Find the equation of the straight line PQ in each of the following diagrams. y

(a)

O

43.

y

(b)

P (6,3)

2

42.

x y − = 1, 2 6

Q

A

C

B

4

Q

x

P (−4,0)

O

x

A straight line L passes through A(2,−3) and is parallel to the line joining the points P ( 2,0) and Q ( −6, 4) . Find the equation of L in (a)

the general form,

(b)

the gradient form,

(c)

the intercept form.

The line 3 x − y + 5 = 0 passes through the point (2, h) and is parallel to the line 4 y = 9 − ( h + k ) x . Find the values of h and k.

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y

44.

R S (−4,6) Q (4,2) x

O

P

In the diagram above, PQRS is a parallelogram. Find

45.

46.

47.

(a)

the gradient of RS,

(b)

the equation of line RS.

Determine whether the two straight lines in each of the following are perpendicular to each other. (a)

Two straight lines that pass through the points A(2,1) and B (5,0) and the points P (2,−1) and Q (3,2) respectively.

(b)

Two straight lines that pass through the points C (1,2) and D (−1,−2) and the points W (7,1) and Z (5,5) respectively.

(c)

2 y = −6 x + 5 and y =

(d)

x y − = 1 and 3 y = − x + 6 2 4

1 x−4. 3

Find the value of the unknown k, given the two straight lines below are perpendicular to each other. 1 x − 1, 6

(a)

2 x + ky = 1, y =

(b)

1 kx + 2 y = 5, 4 x + 3 y = 6 . 2

The straight line tx − 6 y + 7 = 0 is perpendicular to the straight line 4 x + 10 y − 11 = 0 . Calculate the value of t.

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48.

49.

Find the equation of a straight line that passes through (a)

a point P(3, 1) and is perpendicular to a straight line y = 3 x − 4 ,

(b)

a point T ( 4,−1) and is perpendicular to a straight line 2 y − 3 x = 9 ,

(c)

a point V (−1,−2) and is perpendicular to a straight line

x y + = 1. 3 4

Find the equation of the straight line RS in each of the following diagram. y

(a)

y

(b)

R (0,6)

Q(0,5)

R (2,2)

S

O

x

P (−3,0) O

S

x

50.

Given the points E (5,5) and G (7,1) . Find the equation of the perpendicular bisector of EG.

51.

The coordinates of the points P, Q, R and S are (1, 1), (4, 6), (3,−1) and (k , h) respectively. Given that PQRS is a parallelogram, find

52.

(a)

the value of k and of h,

(b)

the area of the parallelogram PQRS.

In diagram below, PQ is perpendicular to QR. y R P (0, h) Q(t ,3)

x

O

Given that the equation of PQ is y = 5 − 2 x , find (a)

the value of h and of t,

(b)

the equation of the straight line QR.

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6.6 EQUATION OF LOCUS Learning Outcomes







Find the equation of the locus that satisfies the following conditions.



The distance of a moving point from a fixed point is constant.



The ratio of the distances of a moving point from two fixed points is constant.

Solving problem involving loci.

Exercise 7 53.

Find the equation of the locus that is always 3 units from a fixed point (0, 4).

54.

P is a moving point such that its distance from a fixed point (1,−1) is always 5 units. Find the equation of locus P.

55.

A point P moves such that its distances from the points A(0, 2) and B( −1, 3) is in the ratio of 2 : 1. Find the equation of locus P.

56.

A point moves such that its distance from the point (2, 0) is three times its distance from the point (4, 0). Find the equation of the locus of the point.

57.

T and U are two points with coordinates of (5,−2) and (−1, 3) respectively. V is a moving point such that 2TV = 3TU . Find the equation of the locus V.

58.

A point P moves so that its distance from two fixed points Q (1, 3) and R( 4,−2) are such that PQ = 2 PR . (a)

Find the equation of the locus of P.

(b)

Determine whether the point (2, 1) lies on the locus of P.

59.

Given the point F (2, 5) and point G (4,−1) . A point N ( x, y ) moves so that ∠FNG is a right angle. Find the equation of the locus of N.

60.

A point moves such that the sum of its distances from the origin and the y-axis is 3 units. Find the equation of the locus of the point.

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