📖 Hover (or tap) any dotted word for its definition. Write your answers in the squared boxes with a pencil, finger or mouse — they save on your device.
(a)The lines l and k have the following equations:
l: 3x − 2y = 0
k: y = −32x + 7
k: y = −32x + 7
Are the lines l and k perpendicular? Justify your answer using calculations.
(a) — answer & justification
(b)A line has equation y = mx + 8, where m ∈ ℝ is a constant. Find the ranges of values of m for which this line does not intersect the line segment from (3, 7) to (3, 11). It may be helpful to draw a diagram.
(b) — working & diagram
(a)p ∈ ℝ is a constant. The point (p, 5) lies on the line 3x − 2y + 28 = 0. Find the value of p.
(a)
(b)The line l has equation y = −13x + 11. The line h has equation 2x − 5y + 10 = 0. Work out the size of the acute angle between the lines l and h. Give your answer correct to the nearest degree.
(b)
(c)A line cuts the x-axis at the point A(a, 0) and the y-axis at B(0, b), where a, b ∈ ℤ. The slope of this line is −23. The area of the triangle enclosed by this line, the x-axis, and the y-axis is 12 square units. There are two different lines that satisfy these conditions. Find the equation of each of these lines. It may be useful to draw a diagram.
(c)
(a)[AB] is a line segment. The point C (6, 11) divides the line segment [AB] internally in the ratio 1 : 3. A is the point (1, 13). Find the co-ordinates of the point B.
(a)
(b)Find the perpendicular distance from the point (5, −2) to the line y = 43 x − 11.
(b)
Ameena, Petro, and Fiadh are taking part in an adventure race. Each unit on the diagram represents 100 metres. The line segment w represents a road, where w has the equation x − 3y = 9 for 0 ≤ y ≤ 8.
(c)Find the co-ordinates of the point on the road w that is closest to the point P(10, 6). It might be useful to find the equation of the line through P that is perpendicular to w.
(c)
(d)Fiadh is at the point F(9, 0) on the road w. She travels 1200 m along the road w away from the point F, in the first quadrant, and then stops. Work out the co-ordinates of the point at which she stops. Give each value correct to 1 decimal place. Remember that each unit on the diagram represents 100 metres.
(d)
(a)Find the area of the triangle with vertices (4, 6), (−3, −1), and (0, 11).
(a)
(b)A(−1, k) and B(5, l) are two points, where k, l ∈ ℚ.
(i)Show that the midpoint of [AB] is (2, k + l2).
(b)(i)
(ii)The perpendicular bisector of [AB] is 3x + 2y − 14 = 0. Find the value of l and the value of k.
(b)(ii)
(a)The points A(8, −4) and B(−1, 3) are the endpoints of the line segment [AB]. Find the coordinates of the point C, which divides [AB] internally in the ratio 4 : 1.
(a)
(b)The line l has a slope of m and contains the point (q, r), where m, q, r ∈ ℝ are all positive. Find the co-ordinates of the point where l cuts the y-axis, in terms of m, q, and r.
(b)
(c)The line k has a slope of −2. The line j makes an angle of 30° with k. Find one possible value of the slope of the line j. Give your answer in the form d + e√f, where d, e, f ∈ ℤ.
(c)
(a)The line 3x − 6y + 2 = 0 contains the point (k, 2k + 23), where k ∈ ℝ. Find the value of k.
(a)
(b)The point P(s, t) is on the line x − 2y − 8 = 0. The point P is also a distance of 1 unit from the line 4x + 3y + 6 = 0. Find a value of s and the corresponding value of t.
(b)
(c)The points A(4, 2) and C(16, 11) are vertices of the triangle ABC (shown). D and E are points on [CA] and [CB]. The ratio |AD| : |DC| is 2 : 1.
(i)Find |AD|.
(c)(i)
(ii)[AB] and [DE] are horizontal line segments. |AB| = 33 units. Find the coordinates of B and of E.
(c)(ii)
(a)The coordinates of three points are A(2, −6), B(6, −12), and C(−4, 3). Find the perpendicular distance from A to BC. Based on your answer, what can you conclude about the relationship between the points A, B, and C?
(a)
(b)The diagram shows two lines a and b. The equation of a is x − 2y + 1 = 0. The acute angle between a and b is θ. Line b makes an angle of 60° with the positive sense of the x-axis (shown). Find the value of θ, in degrees, correct to 3 decimal places.
(b)
(a)The line p makes an intercept on the x-axis at (a, 0) and on the y-axis at (0, b), where a, b ≠ 0. Show that the equation of p can be written as xa + yb = 1.
(a)
(b)(i)The line l has a slope m, and contains the point A(6, 0). Write the equation of the line l in terms of m.
(b)(i)
(ii)The line l cuts the line k: 4x + 3y = 25 at P. Find the co-ordinates of P in terms of m. Give each co-ordinate as a fraction in its simplest form.
(b)(ii)
The line m: 2x + 3y + 1 = 0 is parallel to the line n: 2x + 3y − 51 = 0.
(a)Verify that A(−2, 1) is on m.
(a)
(b)Find the coordinates of B, the point on the line n closest to A, as shown.
(b)
ABC is a triangle where A(0, 6) and C(4, 2). G(23, 43) is the centroid of the triangle. AG intersects BC at P, with |AG| : |GP| = 2 : 1.
(a)Find the co-ordinates of P.
(a)
(b)Find the co-ordinates of B.
(b)
(c)Prove that C is the orthocentre of the triangle ABC.
(c)
The points A(6, −2), B(5, 3) and C(−3, 4) are shown.
(a)Find the equation of the line through B which is perpendicular to AC.
(a)
(b)Use your answer to part (a) to find the co-ordinates of the orthocentre of the triangle ABC.
(b)
A point X has co-ordinates (−1, 6) and the slope of the line XC is 17.
(a)Find the equation of XC. Give your answer in the form ax + by + c = 0, where a, b, c ∈ ℤ.
(a)
The co-ordinates of two points are A(4, −1) and B(7, t). The line l1 : 3x − 4y − 12 = 0 is perpendicular to AB.
(a)Find the value of t.
(a)
(b)Find, in terms of k, the distance between the point P(10, k) and l1.
(b)
(c)(i)P(10, k) is on a bisector of the angles between the lines l1 and l2 : 5x + 12y − 20 = 0. Find the possible values of k.
(c)(i)
(ii)If k > 0, find the distance from P to l1.
(c)(ii)
The line RS cuts the x-axis at the point R and the y-axis at the point S(0, 10). The area of the triangle ROS, where O is the origin, is 1253.
(a)Find the co-ordinates of R.
(a)
(b)Show that the point E(5, 4) is on the line RS.
(b)
(c)A second line y = mx + c, where m and c are positive constants, passes through the point E and again makes a triangle of area 1253 with the axes. Find the value of m and the value of c.
(c)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.