OCR MEI C1 (Core Mathematics 1)

1 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will receive no marks.
\(A\) and \(B\) are points with coordinates \(( - 1,4 )\) and \(( 7,8 )\) respectively.

1. Find the coordinates of the midpoint, M , of AB . Show also that the equation of the perpendicular bisector of AB is \(y + 2 x = 12\).
2. Find the area of the triangle bounded by the perpendicular bisector, the \(y\)-axis and the line AM , as sketched in Fig. 12. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{13979d37-ea09-4d51-aff8-81fa611cc080-1_449_873_843_856} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure}

2 A line has equation \(3 x + 2 y = 6\). Find the equation of the line parallel to this which passes through the point \(( 2,10 )\).

3 Find the coordinates of the point of intersection of the lines \(y = 3 x + 1\) and \(x + 3 y = 6\). \begin{figure}[h] \end{figure} The line AB has equation \(y = 4 x - 5\) and passes through the point \(\mathrm { B } ( 2,3 )\), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C . Find the equation of the line BC and the \(x\)-coordinate of C .
\(5 \mathrm {~A} ( 9,8 ) , \mathrm { B } ( 5,0 )\) an \(\mathrm { C } ( 3,1 )\) are three points.

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