Coordinate Geometry Revision Worksheet

Mr. J GallagherName: ______

Coordinate Geometry Revision Worksheet

Leaving Cert - Ordinary Level

Syllabus

You should be able to:

Explain / understand the following key words

Origin
Distance
Midpoint
Slope
Equation of line
x-Intercept / y-intercept
Equation of a circle
Centre / Radius
Intersection of line & circle

Plot points

Calculate the following

distance
midpoint
slope
equation of line
area of a triangle
equation of a circle
centre & radius of a circle

Show a point is on the line

Graphing two lines to find the point of intersection.

Finding the intercepts and the slope using the equation of the form
y=mx + c

Use the slopes to show two lines are parallel / perpendicular

Solving problems involing the slope of a line.

Solving problems involving a line and a circle.

Distance

a)The centre of a circle is (4,3) and (-3,1) is a point on the circle. Find the length of the radius of the circle.

b)Three points of a triangle are (2,1), (6,7) and (10,1). Investigate is this an equilateral triangle.

c)P = (a,4), Q = (2,3), R = (3,-1) and S = (-2,4) are four points.
If |PQ| = |RS|, find the possible values of a.

d)The distance between (-1,2) and (3,k) is 5. Find the two possible values of k.

Midpoint

a)Find the midpoint of the line segment joining these points:
(i) (2,4) and (0, -1)
(ii) (1,-3) and (2,2)
On which quadrant do each of these midpoints lie?

Slope

a)Find the slope of each of the following sets of points:
(i) (2,4) and (0, -1)
(ii) (1,-3) and (2,2)
(iii) (-3,2) and (-5,0)

b)Line l contains the points (1,1) and (2,4).
Line k contains the points (4,1) and (3,-2).
Investigate if these two lines are parallel.

c)A = (-1,1), B= (1,3), C = (6,2) and D = (4,4) are four points in the plane.
Verify that AB CD.
(i.e. verify both of the slopes are perpendicular.)

d)The line k contains the points (3,-1) and (4,-2).
Find the slope of any line parallel to k.
Find the slope of any line perpendicular to k.

e)The slope of the line through the points (3,2) and (8,k) is ⅗, find the value of k.

f)The slope of the line through (3,-2) and (1,k) is ⅓. Find the value of k.

Equation of a Line

a)Find the equations of the lines through the following pairs of points:
(i) (2,3) and (4,6)
(ii) (-5,1) and (1,0)
(iii) (-3,-5) and (-1,-1)
Hint: find the slope first then sub values into equation of a line formula.

b)Write down the slope of the following line l: 2x + 3y – 7 = 0.
Hence calculate the slope of a line perpendicular to l.
Hint: use to find the slope when given a line.

c)Show that the following lines 2x + 3y – 4 = 0 and 3x – 2y + 7 = 0 are perpendicular.

d)Show that the following lines x – 3y + 4 = 0 and 2x – 6y – 5 = 0 are parallel.

e)If the line x + 3y – 6 = 0 is parallel to the line 2x + ky – 5 = 0, find the value of k.

f)If the line 2x + 3y – 6 = 0 is perpendicular to the line 3x – ky – 5 = 0, find the value of k.

Parallel & Perpendicular Lines

a)Find the equation of a line through the point (4, -1) which is parallel to the line 2x + 3y – 2 = 0.

b)Find the equation of a line through the point (2, -3) which is parallel to the line 2x – y – 2 = 0.

c)Find the equation of a line through the point (4, -1) which is perpendicular to the line 2x + 3y – 2 = 0

d)Find the equation of a line through the point (2, -3) which is parallel to the line 2x – y – 2 = 0

e)The line h contains the points (6,-2) and (-4,10). The equation of the line k is ax + 6y + 12 = 0 and k is perpendicular to h.
Find the value of a.

Point on a Line

a)Investigate is the point (-2, 3) is on the line x + 2y – 4 = 0.

b)Investigate is the point (2, 3) is on the line 3x – 2y + 4 = 0.

c)Verify that (2, -5) is on the line 2x + y + 1 =0.

d)Find the value of k if the line 2x + ky – 8 = 0 contains the point (3, 1).

e)Find the value of k if the line 3x + ky – 3 = 0 contains the point (-1, 2).

f)If (1, 4) is on the line 2x + y + k =0, find the value of k.

Graphing Lines & Intersection of Lines

a)Find the coordinates of the points at which the line l: x – 2y – 6 = 0 intersects the x-axis and y-axis. Use these two points to draw a sketch of the line.
Also find the points which the line k: 2x + 3y – 2 = 0 intersects the x and y-axis. Draw a sketch of k and hence find the point of intersection between the two lines.

b)Use both the graphical methods and the algebraic methods to find the point of intersection between the following pairs of simultaneous equations:
(i) x + y = 5(ii) x + 3y = 7(iii) 2x + 5y = 1
2x – y = 1 2x – y = - 7 x – 3y = - 5
Hence verify your answers by substitution.

Area of a Triangle

a)(2,3), (-5, 1) and (3,1) are the vertices of a triangle. Find the area of the triangle.

b)A = (0, 0), B = (4, -1), C = (2, 3) and D = (-2, 4) are the vertices of a quadrilateral. Find the area of the quadrilateral by dividing it into the two triangles ABC and ACD.

c)(0,0), (4,3) and (6,k) are the vertices of a triangle. Find the value of k if the area of the given triangle is 7 square units.

2011 Exam Question (25 Marks)

2012 Exam Question (25 Marks)

2013 Exam Question (25 Marks)

2014 Exam Question (25 Marks)

2015 Exam Question (25 Marks)

The Circle

Calculating Centre & Radius

a)Write down the centre and radius of the following circles.
(i) x2 + y2 = 9(ii) 4x2 + 4y2 = 9
(iii)x2 + y2 = 27(iv) 9x2 + 9y2 = 25

b)Find the equations of the following circles, given the centre and radius:
(i)centre = (1, 3); radius = 2(ii)centre = (1, -4); radius = 5
(iii) centre = (-2, -3); radius = 6(iv) centre = (0, -2); radius = 2√2

c)Find the centre and radius of the following circles:
(i) (x – 3)2 + (y – 2)2 = 16(ii) (x + 2)2 + (y – 6)2 = 8
(iii) (x – 3)2 + y2 = 5(iv) x2 + (y + 2)2 = 10

Point is On, Outside or Inside circle

a)Show that the point (3, -1) is on the circle x2 + y2 = 10.

b)Show that the point (5, -1) is outside the circle x2 + y2 = 20.

c)Show that the point (1, 2) is on the circle x2 + y2 = 8.

d)Investigate if the point (3, 2) is inside, outside or on the circle x2 + y2 = 10.

e)Show that the point (3, 2) is on the circle (x – 6)2 + (y – 6)2 = 25.

f)Investigate if the point (3, 2) is inside, outside or on the circle
(x – 2)2 + (y + 1)2 = 4

Intersection of Lines & Circles

a)Show that the line l is a tangent to the circle k in each of the following by finding the point of intersection in each case.
(i) l: x + y – 2 = 0 k: x2 + y2 = 2
(ii) l: 2x – y – 5 = 0 k: x2 + y2 = 5
(iii) l: x + 3y – 10 = 0 k: x2 + y2 = 20

A Circle Intersecting the Axis

a)Find the coordinates where the following circles intersection both the x and y-axis.
(i) x2 + y2 = 9(ii)x2 + y2 = 27
(iii) (x – 3)2 + (y – 2)2 = 16(iv) (x + 2)2 + (y – 6)2 = 8