OCR MEI C1 (Core Mathematics 1)

Find the coordinates of the points of intersection of the circle \(x^2 + y^2 = 25\) and the line \(y = 3x\). Give your answers in surd form. [5]

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]

A circle has equation \(x^2 + y^2 = 45\).

1. State the centre and radius of this circle. [2]
2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B. Show that the distance AB is \(\sqrt{162}\). [8]

\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]

1. Points A and B have coordinates \((-2, 1)\) and \((3, 4)\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5x + 3y = 10\). [6]
2. Points C and D have coordinates \((-5, 4)\) and \((3, 6)\) respectively. The line through C and D has equation \(4y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB. Find the coordinates of point E. [3]
3. Find the equation of the circle with centre E which passes through A and B. Show also that CD is a diameter of this circle. [5]

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]