1.03b Straight lines: parallel and perpendicular relationships

A\((9, 8)\), B\((5, 0)\) and C\((3, 1)\) are three points.

1. Show that AB and BC are perpendicular. [3]
2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle. [6]
3. BD is a diameter of the circle. Find the coordinates of D. [3]

\includegraphics{figure_1} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).

1. Find the equation of the line through A and B. [3]
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac{32}{3}\) square units. [2]
3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle. [4]

The points A \((-1, 6)\), B \((1, 0)\) and C \((13, 4)\) are joined by straight lines.

1. Prove that the lines AB and BC are perpendicular. [3]
2. Find the area of triangle ABC. [3]
3. A circle passes through the points A, B and C. Justify the statement that AC is a diameter of this circle. Find the equation of this circle. [6]
4. Find the coordinates of the point on this circle that is furthest from B. [1]

A line \(L\) is parallel to \(y = 4x + 5\) and passes through the point \((-1, 6)\). Find the equation of the line \(L\) in the form \(y = ax + b\). Find also the coordinates of its intersections with the axes. [5]

Find the equation of the line which is perpendicular to the line \(y = 2x - 5\) and which passes through the point \((4, 1)\). Give your answer in the form \(y = ax + b\). [3]

\includegraphics{figure_8} Fig. 10 is a sketch of quadrilateral ABCD with vertices A \((1, 5)\), B \((-1, 1)\), C \((3, -1)\) and D \((11, 5)\).

1. Show that AB = BC. [3]
2. Show that the diagonals AC and BD are perpendicular. [3]
3. Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]

Find the equation of the line which is perpendicular to the line \(y = 5x + 2\) and which passes through the point \((1, 6)\). Give your answer in the form \(y = ax + b\). [3]

Find, in the form \(y = ax + b\), the equation of the line through \((3, 10)\) which is parallel to \(y = 2x + 7\). [3]

The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.

1. Show that \(\angle PQR = 90°\). [4]

Given that \(P\), \(Q\) and \(R\) all lie on circle \(C\),

1. find the coordinates of the centre of \(C\), [3]
2. show that the equation of \(C\) can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]

A ball is projected from a point \(O\) on horizontal ground with speed \(14 \text{ m s}^{-1}\) at an angle of elevation \(30°\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45°\) to the horizontal. The point \(O\) is \(6\) metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O\), \(P\) and \(Q\) lie in the same vertical plane. The line \(PQ\) is a line of greatest slope of the inclined plane. \includegraphics{figure_3}

1. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x\frac{\sqrt{3}}{3} - \frac{x^2}{30}$$ Use \(\cos 30° = \frac{\sqrt{3}}{2}\) and \(\tan 30° = \frac{\sqrt{3}}{3}\). [5 marks]
2. Find the distance \(PQ\). [7 marks]

Point \(C\) has coordinates \((c, 2)\) and point \(D\) has coordinates \((6, d)\). The line \(y + 4x = 11\) is the perpendicular bisector of \(CD\). Find \(c\) and \(d\). [5 marks]

A curve cuts the \(x\)-axis at \((2, 0)\) and has gradient function $$\frac{dy}{dx} = \frac{24}{x^3}$$

1. Find the equation of the curve. [4 marks]
2. Show that the perpendicular bisector of the line joining \(A(-2, 8)\) to \(B(-6, -4)\) is the normal to the curve at \((2, 0)\) [6 marks]

\(ABCD\) is a trapezium where \(A\) is the point \((1, -2)\), \(B\) is the point \((7, 1)\) and \(C\) is the point \((3, 4)\) \(DC\) is parallel to \(AB\). \(AD\) is perpendicular to \(AB\).

1. 1. Find the equation of the line \(CD\). [2 marks]
   2. Show that point \(D\) has coordinates \((-1, 2)\) [3 marks]
2. 1. Find the sum of the length of \(AB\) and the length of \(CD\) in simplified surd form. [2 marks]
   2. Hence, find the area of the trapezium \(ABCD\). [2 marks]

Determine whether the line with equation \(2x + 3y + 4 = 0\) is parallel to the line through the points with coordinates \((9, 4)\) and \((3, 8)\). [4 marks]

Points \(A(-7, -7)\), \(B(8, -1)\), \(C(4, 9)\) and \(D(-11, 3)\) are the vertices of a quadrilateral \(ABCD\).

1. Prove that \(ABCD\) is a rectangle. [4 marks]
2. Find the area of \(ABCD\). [2 marks]

The line \(L_1\) has equation \(x + 7y - 41 = 0\) \(L_1\) is a tangent to the circle C at the point P(6, 5) The line \(L_2\) has equation \(y = x + 3\) \(L_2\) is a tangent to the circle C at the point Q(0, 3) The lines \(L_1\) and \(L_2\) and the circle C are shown in the diagram below. \includegraphics{figure_11}

1. Show that the equation of the radius of the circle through P is \(y = 7x - 37\) [3 marks]
2. Find the equation of C [4 marks]

Point \(A\) has coordinates \((4, 1)\) and point \(B\) has coordinates \((-8, 5)\)

1. Find the equation of the perpendicular bisector of \(AB\) [5 marks]
2. A circle passes through the points \(A\) and \(B\) A diameter of the circle lies along the \(x\)-axis. Find the equation of the circle. [4 marks]

The point \(A\) has coordinates \((-1, a)\) and the point \(B\) has coordinates \((3, b)\) The line \(AB\) has equation \(5x + 4y = 17\) Find the equation of the perpendicular bisector of the points \(A\) and \(B\). [4 marks]

The line \(L\) has equation \(2x + 3y = 7\) Which one of the following is perpendicular to \(L\)? Tick one box. [1 mark] \(2x - 3y = 7\) \(3x + 2y = -7\) \(2x + 3y = -\frac{1}{7}\) \(3x - 2y = 7\)