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

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]

A is the point \((1, 5)\) and B is the point \((6, -1)\). M is the midpoint of AB. Determine whether the line with equation \(y = 2x - 5\) passes through M. [3]

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]

Find the equation of the line with gradient \(-2\) which passes through the point \((3, 1)\). Give your answer in the form \(y = ax + b\). Find also the points of intersection of this line with the axes. [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 the equation of the line passing through \((-1, -9)\) and \((3, 11)\). Give your answer in the form \(y = mx + c\). [3]

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

A circle \(C\) has centre \((3, 4)\) and radius \(3\sqrt{2}\). A straight line \(l\) has equation \(y = x + 3\).

1. Write down an equation of the circle \(C\). [2]
2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds. [5]
3. Find the distance between these two points. [2]

The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P\) (ln 2, \(p + 2q\)) on \(C\), the gradient is 5.

1. Find the value of \(p\) and the value of \(q\). [5]

The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).

1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately 53.8 [5]

A curve has parametric equations $$x = t(t - 1), \quad y = \frac{4t}{1-t}, \quad t \neq 1.$$

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [4]

The point \(P\) on the curve has parameter \(t = -1\).

1. Show that the tangent to the curve at \(P\) has the equation $$x + 3y + 4 = 0.$$ [3]

The tangent to the curve at \(P\) meets the curve again at the point \(Q\).

1. Find the coordinates of \(Q\). [7]

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]

In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curve with equation \(4xy = 2(x^2 + 4y^2) - 9x\).

1. Show that \(\frac{dy}{dx} = \frac{4x - 4y - 9}{4x - 16y}\). [3] At the point \(P\) on the curve the tangent to the curve is parallel to the \(y\)-axis and at the point \(Q\) on the curve the tangent to the curve is parallel to the \(x\)-axis.
2. Show that the distance \(PQ\) is \(k\sqrt{5}\), where \(k\) is a rational number to be determined. [8]

Three of the following points lie on the same straight line. Which point does not lie on this line? Tick one box. [1 mark] \((-2, 14)\) \((-1, 8)\) \((1, -1)\) \((2, -6)\)

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]

Line \(L\) has equation $$5y = 4x + 6$$ Find the gradient of a line parallel to line \(L\) Circle your answer. $$\frac{5}{4} \quad -\frac{4}{5} \quad \frac{4}{5} \quad \frac{5}{4}$$ [1 mark]

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]