1.03b Straight lines: parallel and perpendicular relationships

\includegraphics{figure_3} The curve \(C_1\), shown in Figure 3, has equation \(y = 4x^2 - 6x + 4\). The point \(P\left(\frac{1}{2}, 2\right)\) lies on \(C_1\) The curve \(C_2\), also shown in Figure 3, has equation \(y = \frac{1}{2}x + \ln(2x)\). The normal to \(C_1\) at the point \(P\) meets \(C_2\) at the point \(Q\). Find the exact coordinates of \(Q\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [8]

A circle \(C\) with centre at \((-2, 6)\) passes through the point \((10, 11)\).

1. Show that the circle \(C\) also passes through the point \((10, 1)\). [3]

The tangent to the circle \(C\) at the point \((10, 11)\) meets the \(y\) axis at the point \(P\) and the tangent to the circle \(C\) at the point \((10, 1)\) meets the \(y\) axis at the point \(Q\).

1. Show that the distance \(PQ\) is \(58\) explaining your method clearly. [7]

The line \(l_1\) has equation \(2x - 3y = 9\) The line \(l_2\) passes through the points \((3, -1)\) and \((-1, 5)\) Determine, giving full reasons for your answer, whether lines \(l_1\) and \(l_2\) are parallel, perpendicular or neither.

A curve with centre \(C\) has equation $$x^2 + y^2 + 2x - 6y - 40 = 0$$

1. 1. State the coordinates of \(C\).
   2. Find the radius of the circle, giving your answer as \(r = n\sqrt{2}\). [3]
2. The line \(l\) is a tangent to the circle and has gradient \(-7\). Find two possible equations for \(l\), giving your answers in the form \(y = mx + c\). [8]

In this question you must show detailed reasoning. The centre of a circle C is at the point \((-1, 3)\) and C passes through the point \((1, -1)\). The straight line L passes through the points \((1, 9)\) and \((4, 3)\). Show that L is a tangent to C. [7]

The line \(L_1\) passes through the points \(A(-1, 3)\) and \(B(2, 9)\). The line \(L_2\) has equation \(2y + x = 25\) and intersects \(L_1\) at the point \(C\). \(L_2\) also intersects the \(x\)-axis at the point \(D\).

1. Show that the equation of the line \(L_1\) is \(y = 2x + 5\). [3]
2. 1. Find the coordinates of the point \(D\).
   2. Show that \(L_1\) and \(L_2\) are perpendicular.
   3. Determine the coordinates of \(C\). [5]
3. Find the length of \(CD\). [2]
4. Calculate the angle \(ADB\). Give your answer in degrees, correct to one decimal place. [5]

The line \(L_1\) passes through the points \(A(0, 5)\) and \(B(3, -1)\).

1. Find the equation of the line \(L_1\). [3]

The line \(L_2\) is perpendicular to \(L_1\) and passes through the origin \(O\).

1. Write down the equation of \(L_2\). [1]

The lines \(L_1\) and \(L_2\) intersect at the point \(C\).

1. Calculate the area of triangle \(OAC\). [4]
2. Find the equation of the line \(L_3\) which is parallel to \(L_1\) and passes through the point \(D(4, 2)\). [2]
3. The line \(L_3\) intersects the \(y\)-axis at the point \(E\). Find the area of triangle \(ODE\). [1]

The point \(A\) has coordinates \((-2, 5)\) and the point \(B\) has coordinates \((3, 8)\). The point \(C\) lies on the \(x\)-axis such that \(AC\) is perpendicular to \(AB\).

1. Find the equation of \(AB\). [3]
2. Show that \(C\) has coordinates \((1, 0)\). [3]
3. Calculate the area of triangle \(ABC\). [4]
4. Find the equation of the circle which passes through the points \(A\), \(B\) and \(C\). [5]

The points \(A(0, 2)\), \(B(-2, 8)\), \(C(20, 12)\) are the vertices of the triangle \(ABC\). The point \(D\) is the mid-point of \(AB\).

1. Show that \(CD\) is perpendicular to \(AB\). [6]
2. Find the exact value of \(\tan CAB\). [5]
3. Write down the geometrical name for the triangle \(ABC\). [1]

\includegraphics{figure_2} Figure 2 is a sketch showing the line \(l_1\) with equation \(y = 2x - 1\) and the point \(A\) with coordinates \((-2, 3)\). The line \(l_2\) passes through \(A\) and is perpendicular to \(l_1\)

1. Find the equation of \(l_2\) writing your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants to be found. [3]

The point \(B\) and the point \(C\) lie on \(l_1\) such that \(ABC\) is an isosceles triangle with \(AB = AC = 2\sqrt{13}\)

1. Show that the \(x\) coordinates of points \(B\) and \(C\) satisfy the equation $$5x^2 - 12x - 32 = 0$$ [4]

Given that \(B\) lies in the 3rd quadrant

1. find, using algebra and showing your working, the coordinates of \(B\). [4]

\includegraphics{figure_7} The diagram shows a circle which passes through the points \(A(2, 9)\) and \(B(10, 3)\). \(AB\) is a diameter of the circle.

1. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the exact coordinates of \(C\). [4]
2. Find the exact area of the triangle formed by \(B\), \(C\) and the centre of the circle [3]

In this question you must show detailed reasoning. The centre of a circle C is the point (-1, 3) and C passes through the point (1, -1). The straight line L passes through the points (1, 9) and (4, 3). Show that L is a tangent to C. [7]

The trapezium \(ABCD\) is shown below. \includegraphics{figure_2} The line \(AB\) has equation \(2x + 3y = 14\) and \(DC\) is parallel to \(AB\). The point D has coordinates \((3, 7)\).

1. Find an equation of the line DC [2 marks]
2. The angle BAD is a right angle. Find an equation of the line AD, giving your answer in the form \(mx + ny + p = 0\), where \(m\), \(n\) and \(p\) are integers. [3 marks]

\includegraphics{figure_2} The figure above shows a triangle with vertices at \(A(2,6)\), \(B(11,6)\) and \(C(p,q)\).

1. Given that the point \(D(6,2)\) is the midpoint of \(AC\), determine the value of \(p\) and the value of \(q\). [2]

The straight line \(l\) passes through \(D\) and is perpendicular to \(AC\). The point \(E\) is the intersection of \(l\) and \(AB\).

1. Find the coordinates of \(E\). [4]

The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_7}

1. Determine the coordinates of \(B\). [6]
2. Hence determine the exact value of \(\tan\alpha\). [3]

\includegraphics{figure_5} Figure 4 The line \(l_1\) has equation \(y = \frac{3}{5}x + 6\) The line \(l_2\) is perpendicular to \(l_1\) and passes through the point \(B(8, 0)\), as shown in the sketch in Figure 4.

1. Show that an equation for line \(l_2\) is $$5x + 3y = 40$$ [3]

Given that

• lines \(l_1\) and \(l_2\) intersect at the point C
• line \(l_1\) crosses the \(x\)-axis at the point A

1. find the exact area of triangle \(ABC\), giving your answer as a fully simplified fraction in the form \(\frac{p}{q}\) [5]

A line has equation \(y = 2x\) and a circle has equation \(x^2 + y^2 + 2x - 16y + 56 = 0\).

1. Show that the line does not meet the circle. [3]
2. 1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2x\). [4]
   2. Hence find the shortest distance between the line \(y = 2x\) and the circle, giving your answer in an exact form. [4]

Determine the equation of the line that passes through the point \((1, 3)\) and is perpendicular to the line with equation \(3x + 6y - 5 = 0\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]

1. Find the shortest distance between the point \((-6, 4)\) and the line \(y = -0.75x + 7\). [2]

Two lines, \(l_1\) and \(l_2\), are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$$

1. Find the shortest distance between \(l_1\) and \(l_2\). [3]
2. Hence determine the geometrical arrangement of \(l_1\) and \(l_2\). [2]

A curve has equation \(y = kx^{\frac{1}{2}}\) where \(k\) is a constant. The point \(P\) on the curve has \(x\)-coordinate 4. The normal to the curve at \(P\) is parallel to the line \(2x + 3y = 0\) and meets the \(x\)-axis at the point \(Q\). The line \(PQ\) is the radius of a circle centre \(P\). Show that \(k = \frac{1}{2}\). Find the equation of the circle. [10]

\includegraphics{figure_2} The diagram shows a triangle \(ABC\). The vertices have coordinates \(A(3, -7)\), \(B(9, 1)\) and \(C(-1, -5)\).

1. 1. Find the length of the side \(AB\). [2]
   2. Find the coordinates of the mid-point of \(AB\). [1]
   3. A circle has diameter \(AB\). Find the equation of the circle in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\), \(b\) and \(r\) are constants to be found. [3]
2. Find the equation of the line \(l\) passing through \(B\) parallel to \(AC\). [3]