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IB Mathematics: Applications and Interpretation SL · 12.3: Coordinate Geometry · 12.3.2

Name Date

Straight lines: gradient, parallel and perpendicular Problem set B

Show the gradients and the substitution. Use the calculator to solve pairs of equations. Give 3 significant figures unless the answer is exact.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A line has equation \(3x + 4y - 24 = 0\).

    1. (a) Write down its gradient and its y-intercept.
    2. (b) Find the x-intercept.
    3. (c) Write the equation in the form \(y = mx + c\).

  2. Find the equation of the line parallel to \(3x - 4y = 12\) that passes through \((-4, 2)\). Give it in the form \(ax + by + d = 0\) with whole-number coefficients.

  3. For each pair of lines, say whether they are parallel, perpendicular or neither.

    1. (a) \(y = -2x + 7\) and \(4x + 2y - 3 = 0\)
    2. (b) \(5x + 2y = 10\) and \(2x - 5y = 3\)
    3. (c) \(y = 2x + 1\) and \(y = -x + 5\)

  4. The lines \(kx + 3y = 7\) and \(5x - 2y = 3\) have gradients \(m_1\) and \(m_2\).

    1. (a) Find the value of k for which the lines are parallel.
    2. (b) Find the value of k for which the lines are perpendicular.

  5. The points \(P(-4, -3)\), \(Q(0, 3)\) and \(R(2, 6)\) are given.

    1. (a) Show that P, Q and R lie on a straight line.
    2. (b) The point \(S(10, t)\) is on the same line. Find t.

In context

  1. A plumber charges a call-out fee plus the same amount for each hour of work. A job of 3 hours costs €135 and a job of 6 hours costs €240. Treat the cost as a linear function of the number of hours.

    1. (a) Find the gradient of the line through these two points and say what it means.
    2. (b) Find the intercept and say what it means.
    3. (c) Find the cost after 8 hours.
    4. (d) The cost is €380. Find the number of hours.

  2. On a plan a straight cycle path passes through \(M(-2, 4)\) and \(N(6, 0)\). A new straight path is to be built through \(T(3, 10)\), parallel to MN. Distances are in metres.

    1. (a) Find the gradient of the path MN.
    2. (b) Find the equation of the new path. Give it in the form \(y = mx + c\).
    3. (c) Find where the new path meets the y-axis.

  3. A triangular sail has corners \(P(1, 2)\), \(Q(5, 5)\) and \(R(11, -3)\), with coordinates in centimetres.

    PQR
    1. (a) Find the gradients of PQ and QR.
    2. (b) Show that angle PQR is a right angle.
    3. (c) Find the area of the sail.

Stretch

  1. The lines \((k+2)x + 3y = 5\) and \(4x + (k-1)y = 9\) have gradients \(m_1\) and \(m_2\).

    1. (a) Write \(m_1\) and \(m_2\) in terms of k.
    2. (b) Find the values of k for which the lines are parallel. Give your answers to 3 significant figures.
    3. (c) Find the value of k for which the lines are perpendicular.

  2. ABCD is a parallelogram. The points \(A(-2, 0)\), \(B(3, -1)\) and \(C(7, 3)\) are three of its vertices, taken in order round the shape.

    ABC
    1. (a) Find the gradients of AB and BC.
    2. (b) Find the equation of the line through A parallel to BC, and the equation of the line through C parallel to AB. Give each in the form \(ax + by + d = 0\).
    3. (c) Solve the two equations together to find the coordinates of D.

jmaths.xyz · AI SL · 12.3.2 Straight lines: gradient, parallel and perpendicular