Examples:
refer pg3 - pg6 for solutions + answers!
(Distance, Midpoint, Gradient)
1. A(-2,1) and B(-6,5) are the opposite ends of the diameter of a circle. Find the length of diameter
and the coordinates of its centre.
2. M(5,7) is the mid-point of the line segment joining A(3,4) to B. Find the coordinates of B.
3. Show that triangle formed by the points (-3,-2), (2,-7) and (-2,5) is isosceles.
4. Find the gradients of the lines AB and BC where A is (3,4), B is (7,6) and C is (-3,1). What can you
deduce about the points A, B and C?
5. The point P(x,y) lies on the straight line joining A(3,0) and B(5,6). Find expression for the
gradients of AP and PB. Hence, show that y=3x-9.
6. The vertices of a quadrilateral PQRS are P(1,2), Q(7,0) R(6,-4) and S(-3,-1).
i. Find the gradient of each side of the quadrilateral
ii. What type of quadrilateral is PQRS?
7. A triangle has vertices A(-2,1), B(3,-4) and C(5,7)
i. Find the coordinates of M, the midpoint of AB, and N the midpoint of AC.
ii. Show that MN is parallel to BC.
(Equation, Point of intersection, intersection of straight line and curve)
8. Find the equations of the straight lines through the given points with gradient shown.
i. (2,3), gradient 5
ii. (0,4) gradient m
iii. (3,0) gradient -3/5
iv. (3,-2) gradient -5/8
9. Find the equations of the lines joining the following pairs of points. Leave your final answer
without fractions and in one of the forms 𝑦 = 𝑚𝑥 + 𝑐 or 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0
i. (10,-3) and (-5,-12)
ii. (-1,0) and (0,-1)
iii. (-5,4) and (-2,-1)
iv. (0,0)and (p,q)
10. Find the equation of the line through (3,9) parallel to the line joining (-3,2) and (2,-3)
11. Find the equation of the line through (4,-3) parallel to y+2x=7
12. Find the equation of the line through the point (-2, 5) which is perpendicular to the line 𝑦 =
3𝑥 + 1. Find also the point of intersection of the two lines.
13. Find the equation of the line through the point (1, 1) which is perpendicular to the line 2𝑥 −
3𝑦 = 12. Find also the point intersection of the two lines.
14. Find the points of intersection of the following pairs of straight lines
i. 3𝑥 + 4𝑦 = 33, 2𝑦 = 𝑥 − 1
ii. 𝑦 = 2𝑥 + 3, 4𝑥 − 2𝑦 = −6
iii. 𝑎𝑥 + 𝑏𝑦 = 𝑐, 𝑦 = 2𝑎𝑥
(Relationship between tangent to a curve and a repeated root of an equation)
15. Find the values of c which the line 𝑦 = 2𝑥 + 𝑐 is a tangent to the circle
𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 − 7 = 0
16. Prove that the line 𝑦 = 3𝑥 + 1 is a tangent to the circle 𝑥 2 + 𝑦 2 − 14𝑥 − 4𝑦 + 13 = 0
Prove that part of the line 𝑦 = 3𝑥 + 5 forms a chord to the circle 𝑥 2 + 𝑦 2 − 2𝑥 − 6𝑦 + 5 = 0
and find the length of this chord.
17. Find the values or ranges of value 𝜆 for which the line 𝑦 = 2𝑥 + 𝜆
a) Touches b) cuts in real points c) does not meet
The curve 2𝑥 2 + 𝑦 2 = 4
,(Perpendicular bisector, Triangles, Rectangle)
18. The diagram 1 shows a kite ABCD with AB=AD and CB=CD. The point A lies on the x-axis, the
point B is (4,6),the point D is (13,9) and the equation of BC is
y=2x-2.
i. The gradient of BD and hence that of AC
ii. Equation of AC
iii. The coordinates of A and of C
iv. The length of BD and of AC
v. The area of the kite ABCD
19. The diagram shows a trapezium ABCD in which BC is parallel to AD and CD is perpendicular to
both BC and AD. The coordinates of A, B and C are (0,2), (3,13) and (12,16) respectively. Find:
i. The equation of AD and CD
ii. The coordinates of D
The line AB produced meets the line DC produced at E. Find
i. The coordinates of E
ii. The ratio of AE:BE
iii. The ratio of the area of the triangle BEC to the area of
the trapezium ABCD.
20. The diagram shows a triangle ABC where A is (6,9), B is (-2,3) and C is (h,-5). Given that AB=BC,
and that h is positive,
i. find the value of h
ii. show that the angle ABC =90𝑜
The midpoint of AB is M. The line through M parallel to
BC meets AC at the point P and the x-axis at the point Q.
Find:
i. The coordinates of M, P and Q
ii. The ratio of MP:PQ
21. Find the coordinates of the foot of the perpendicular
from A(-2,-4) to the line joining B(0,2) and C(-1,4).
,
refer pg3 - pg6 for solutions + answers!
(Distance, Midpoint, Gradient)
1. A(-2,1) and B(-6,5) are the opposite ends of the diameter of a circle. Find the length of diameter
and the coordinates of its centre.
2. M(5,7) is the mid-point of the line segment joining A(3,4) to B. Find the coordinates of B.
3. Show that triangle formed by the points (-3,-2), (2,-7) and (-2,5) is isosceles.
4. Find the gradients of the lines AB and BC where A is (3,4), B is (7,6) and C is (-3,1). What can you
deduce about the points A, B and C?
5. The point P(x,y) lies on the straight line joining A(3,0) and B(5,6). Find expression for the
gradients of AP and PB. Hence, show that y=3x-9.
6. The vertices of a quadrilateral PQRS are P(1,2), Q(7,0) R(6,-4) and S(-3,-1).
i. Find the gradient of each side of the quadrilateral
ii. What type of quadrilateral is PQRS?
7. A triangle has vertices A(-2,1), B(3,-4) and C(5,7)
i. Find the coordinates of M, the midpoint of AB, and N the midpoint of AC.
ii. Show that MN is parallel to BC.
(Equation, Point of intersection, intersection of straight line and curve)
8. Find the equations of the straight lines through the given points with gradient shown.
i. (2,3), gradient 5
ii. (0,4) gradient m
iii. (3,0) gradient -3/5
iv. (3,-2) gradient -5/8
9. Find the equations of the lines joining the following pairs of points. Leave your final answer
without fractions and in one of the forms 𝑦 = 𝑚𝑥 + 𝑐 or 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0
i. (10,-3) and (-5,-12)
ii. (-1,0) and (0,-1)
iii. (-5,4) and (-2,-1)
iv. (0,0)and (p,q)
10. Find the equation of the line through (3,9) parallel to the line joining (-3,2) and (2,-3)
11. Find the equation of the line through (4,-3) parallel to y+2x=7
12. Find the equation of the line through the point (-2, 5) which is perpendicular to the line 𝑦 =
3𝑥 + 1. Find also the point of intersection of the two lines.
13. Find the equation of the line through the point (1, 1) which is perpendicular to the line 2𝑥 −
3𝑦 = 12. Find also the point intersection of the two lines.
14. Find the points of intersection of the following pairs of straight lines
i. 3𝑥 + 4𝑦 = 33, 2𝑦 = 𝑥 − 1
ii. 𝑦 = 2𝑥 + 3, 4𝑥 − 2𝑦 = −6
iii. 𝑎𝑥 + 𝑏𝑦 = 𝑐, 𝑦 = 2𝑎𝑥
(Relationship between tangent to a curve and a repeated root of an equation)
15. Find the values of c which the line 𝑦 = 2𝑥 + 𝑐 is a tangent to the circle
𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 − 7 = 0
16. Prove that the line 𝑦 = 3𝑥 + 1 is a tangent to the circle 𝑥 2 + 𝑦 2 − 14𝑥 − 4𝑦 + 13 = 0
Prove that part of the line 𝑦 = 3𝑥 + 5 forms a chord to the circle 𝑥 2 + 𝑦 2 − 2𝑥 − 6𝑦 + 5 = 0
and find the length of this chord.
17. Find the values or ranges of value 𝜆 for which the line 𝑦 = 2𝑥 + 𝜆
a) Touches b) cuts in real points c) does not meet
The curve 2𝑥 2 + 𝑦 2 = 4
,(Perpendicular bisector, Triangles, Rectangle)
18. The diagram 1 shows a kite ABCD with AB=AD and CB=CD. The point A lies on the x-axis, the
point B is (4,6),the point D is (13,9) and the equation of BC is
y=2x-2.
i. The gradient of BD and hence that of AC
ii. Equation of AC
iii. The coordinates of A and of C
iv. The length of BD and of AC
v. The area of the kite ABCD
19. The diagram shows a trapezium ABCD in which BC is parallel to AD and CD is perpendicular to
both BC and AD. The coordinates of A, B and C are (0,2), (3,13) and (12,16) respectively. Find:
i. The equation of AD and CD
ii. The coordinates of D
The line AB produced meets the line DC produced at E. Find
i. The coordinates of E
ii. The ratio of AE:BE
iii. The ratio of the area of the triangle BEC to the area of
the trapezium ABCD.
20. The diagram shows a triangle ABC where A is (6,9), B is (-2,3) and C is (h,-5). Given that AB=BC,
and that h is positive,
i. find the value of h
ii. show that the angle ABC =90𝑜
The midpoint of AB is M. The line through M parallel to
BC meets AC at the point P and the x-axis at the point Q.
Find:
i. The coordinates of M, P and Q
ii. The ratio of MP:PQ
21. Find the coordinates of the foot of the perpendicular
from A(-2,-4) to the line joining B(0,2) and C(-1,4).
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