1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

Find the gradient of the line with equation \(2x + 5y = 7\) Circle your answer. [1 mark] \(\frac{2}{5}\) \quad \(\frac{5}{2}\) \quad \(-\frac{2}{5}\) \quad \(-\frac{5}{2}\)

The line \(L\) has equation $$5y + 12x = 298$$ A circle, \(C\), has centre \((7, 9)\) \(L\) is a tangent to \(C\).

1. Find the coordinates of the point of intersection of \(L\) and \(C\). Fully justify your answer. [5 marks]
2. Find the equation of \(C\). [3 marks]

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

The line \(l\) passes through the points \(A (3, 1)\) and \(B (4, -2)\). Find an equation for \(l\). [3]

\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]

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.

\includegraphics{figure_4} The value of a sculpture, \(£V\), is modelled by the equation \(V = Ap^t\), where \(A\) and \(p\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1960. The line \(l\) shown in Figure 4 illustrates the linear relationship between \(t\) and \(\log_{10}V\) for \(t \geq 0\). The line \(l\) passes through the point \((0, \log_{10}20)\) and \((50, \log_{10}2000)\).

1. Write down the equation of the line \(l\). [3]
2. Using your answer to part a or otherwise, find the values of \(A\) and \(p\). [4]
3. With reference to the model, interpret the values of the constant \(A\) and \(p\). [2]
4. Use your model, to predict the value of the sculpture, on 1st January 2020, giving your answer to the nearest pounds. [1]

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 points \(A(-2, 4)\) and \(B(6, 10)\) are such that \(AB\) is the diameter of a circle.

1. Show that the centre of the circle has coordinates \((2, 7)\). [1]
2. The equation of the circle is \(x^2 + y^2 + ax + by + c = 0\). Determine the values of \(a\), \(b\), \(c\). [3]

A straight line, with equation \(y = x + 6\), passes through the point \(A\) and cuts the circle again at the point \(C\).

1. Find the coordinates of \(C\). [5]
2. Calculate the exact area of the triangle \(ABC\). [3]

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]

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]

A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5x^2 + (2r - 48)x + (r^2 - 24r + 144) = 0$$ [3]

Given also that \(l\) is a tangent to \(C\),

1. find the two possible values of \(r\), giving your answers as fully simplified surds. [4]

\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]

A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.

1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
2. 1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
   2. Give a geometrical interpretation of the case in part (b)(i). [1]
3. Give a geometrical interpretation of the case where \(a = 5\). [1]

\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 circle, C, has equation \(x^2 - 6x + y^2 = 16\). A second circle, D, has the following properties:

• The line through the centres of circle C and circle D has gradient 1.
• Circle D touches circle C at exactly one point.
• The centre of circle D lies in the first quadrant.
• Circle D has the same radius as circle C.

Find the coordinates of the centre of circle D. [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]

The circle \(x^2 + y^2 + 2x - 14y + 25 = 0\) has its centre at the point C. The line \(7y = x + 25\) intersects the circle at points A and B. Prove that triangle ABC is a right-angled triangle. [7]