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UNIVERSITY OF STRATHCLYDE
DEPARTMENT OF MATHEMATICS & STATISTICS
SUMMER SCHOOL
Exercises on Geometry
1. Find the lengths of the sides of a triangle whose vertices are the points (5, −6), (−3, −2)
and (1, −3) .
2. Show that the points (−2, 2), (2, 3) and (−1, −2) form an isosceles triangle.
3. Find the equation of the straight line which:
(a) passes through (−1, −3) and (5, −1) ;
(b) passes through (1, 2) and makes an angle of 45◦ with the x-axis ;
(c) makes intercepts of −5 and 3 respectively on the x and y axes ;
(d) cuts off an intercept of 3 units from the negative y-axis and makes an angle of
120◦ with the x-axis .
4. Find the acute angle between the lines 2x − y + 5 = 0 and 3x + y − 7 = 0 .
5. Find the coordinates of the point of intersection P of the two straight lines
4x + 3y − 7 = 0 and 3x − 4y + 1 = 0 .
Determine the equation of the straight line joining P to the point (−2, 3) .
6. Find the equation of the straight line which:
(a) is parallel to the y-axis and passes through (2, 3) ;
(b) is parallel to the line 4x + 3y + 8 = 0 and passes through (2, −3) ;
(c) is perpendicular to the line 4x + 3y + 8 = 0 and passes through the origin.
7. Find the equation of the straight line joining the points (3, 6) and (5, 7). Show that
this line is perpendicular to the straight line joining the points (−3, 4) and (−2, 2) .
8. Find the equation of the perpendicular bisector of AB if A is (3, 2) and B is (5, 1) .
9. Find the equations of the two straight lines which make angles of 45◦ with the straight
line 4x + 3y − 21 = 0 and which pass through the point (1, −3) .
10. Find the coordinates of the foot of the perpendicular from the point (5, 7) on the
straight line which joins the points (6, −1) and (1, 6) .
11. Find the centre and radius of each of the following circles:
(a) x2 + y 2 − 5x = 0 ;
(b) 9x2 + 9y 2 + 27x + 12y + 19 = 0 ;
(c) 4x2 + 4y 2 − 28y + 33 = 0 ;
(d) x2 + y 2 − 2x − 6y + 6 = 0 .
12. Find the equation of the circle which passes through the points:
(a) (4, 0), (0, −6) and the origin ;
(b) (2, 1), (−2, 5) and (−3, 2) ;
(c) (6, 1), (3, 2) and (2, 3) .
13. Find the equation of the circle whose centre lies on the x-axis and which passes through
the points (6, 0) and (0, 10) .
14. Find the equation of the tangent at the point (1, 0) on the circle with equation
x2 + y 2 − 5x − y + 4 = 0 .
15. The coordinates of A and B are (−2, 2) and (3, 1) respectively. Show that the
equation of the circle which has AB as diameter is x2 + y 2 − x − 3y − 4 = 0 .
16. The straight line y = x cuts the circle x2 + y 2 + 2x + 6y − 192 = 0 in P and Q.
Find the equation of the circle with P Q as diameter.
17. Prove that the line 3x + 4y − 13 = 0 is a tangent to the circle x2 + y 2 − 2x − 3 = 0 .
18. Show that the point (1, 1) is equidistant from the lines
3x + 4y − 12 = 0, 5x − 12y + 20 = 0 and 4x − 3y − 6 = 0 .
19. Find the equation of the circle, with centre at the given point, which touches the given
line:
(a) (4, −7),
3x + 4y − 9 = 0 ;
(b) (7, −6),
3x − 4y + 5 = 0 ;
(c) (3, −2),
x+y−3=0 .
20. Does the point (1, 2) lie inside the circle x2 + y 2 + 4x + 2y − 11 = 0 ?