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

Name Date

Straight lines: gradient, parallel and perpendicular Problem set A

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

Difficulty: first skills applying them stretch

Skills

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

    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 \(2x + 5y = 10\) that passes through \((5, -3)\). 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) \(y = 3x - 1\) and \(6x - 2y + 5 = 0\)
    2. (b) \(2x + 3y = 6\) and \(3x - 2y = 1\)
    3. (c) \(x + 2y = 4\) and \(y = \frac12 x + 1\)

  4. The lines \(kx + 4y = 9\) and \(2x - 3y = 5\) 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 \(P(-2, 1)\), \(Q(1, 7)\) and \(R(4, 13)\) are given.

    1. (a) Show that P, Q and R lie on a straight line.
    2. (b) The point \(S(5, t)\) is on the same line. Find t.

In context

  1. A gym charges a fixed joining fee plus the same amount each month. A membership of 4 months costs €140 and a membership of 10 months costs €290. Treat the cost as a linear function of the number of months.

    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 14 months.
    4. (d) The cost is €365. Find the number of months.

  2. On a map a straight road passes through \(P(1, 2)\) and \(Q(7, 5)\). A new straight road is to be built through \(T(2, 9)\), parallel to PQ. Distances are in kilometres.

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

  3. A triangular plot of land has corners \(P(-2, 1)\), \(Q(2, 3)\) and \(R(3, 1)\), with coordinates in metres.

    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 plot.

Stretch

  1. The lines \((k+1)x + 2y = 7\) and \(3x + (k-2)y = 4\) 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 \(A(1, 1)\), \(B(6, 3)\) and \(C(9, 8)\) 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 HL · 12.3.2 Straight lines: gradient, parallel and perpendicular