1.03e Complete the square: find centre and radius of circle

A circle \(C\) has equation $$x^2 + y^2 - 6x + 8y - 75 = 0.$$

1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). [3]

A second circle has centre at the point \((15, 12)\) and radius \(10\).

1. Sketch both circles on a single diagram and find the coordinates of the point where they touch. [4]

A circle \(C\) has equation $$x^2 + y^2 - 10x + 6y - 15 = 0.$$

1. Find the coordinates of the centre of \(C\). [2]
2. Find the radius of \(C\). [2]

The point \(A\) has coordinates \((2, 5)\) and the point \(B\) has coordinates \((-2, 8)\). Find, in cartesian form, an equation of the circle with diameter \(AB\). [4]

A circle \(C\) has equation \(x^2 + y^2 - 10x + 6y - 15 = 0\).

1. Find the coordinates of the centre of \(C\). [2]
2. Find the radius of \(C\). [2]

A circle \(C\) has equation \(x^2 + y^2 - 6x + 8y - 75 = 0\).

1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). [3]

A second circle has centre at the point \((15, 12)\) and radius \(10\).

1. Sketch both circles on a single diagram and find the coordinates of the point where they touch. [4]

The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.

1. Show that \(\angle PQR = 90°\). [4]

Given that \(P\), \(Q\) and \(R\) all lie on circle \(C\),

1. find the coordinates of the centre of \(C\), [3]
2. show that the equation of \(C\) can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]

\includegraphics{figure_3} Figure 3 shows the circle \(C\) with equation $$x^2 + y^2 - 8x - 10y + 16 = 0.$$

1. Find the coordinates of the centre and the radius of \(C\). [3]

\(C\) crosses the \(y\)-axis at the points \(P\) and \(Q\).

1. Find the coordinates of \(P\) and \(Q\). [3]

The chord \(PQ\) subtends an angle of \(\theta\) at the centre of \(C\).

1. Using the cosine rule, show that \(\cos \theta = \frac{7}{25}\). [4]
2. Find the area of the shaded minor segment bounded by \(C\) and the chord \(PQ\). [4]

A circle has the equation \(x^2 + y^2 - 6y - 7 = 0\).

1. Find the coordinates of the centre of the circle. [2]
2. Find the radius of the circle. [2]

A circle has the equation $$x^2 + y^2 + 8x - 4y + k = 0,$$ where \(k\) is a constant.

1. Find the coordinates of the centre of the circle. [2]

Given that the \(x\)-axis is a tangent to the circle,

1. find the value of \(k\). [3]

An ellipse \(E\) has equation $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$

1. Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes. [2 marks]
2. Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(-5 < k < 5\). [5 marks]
3. The ellipse \(E\) is translated by the vector \(\begin{bmatrix} a \\ b \end{bmatrix}\) to form another ellipse whose equation is \(9x^2 + 16y^2 + 18x - 64y = c\). Find the values of the constants \(a\), \(b\) and \(c\). [5 marks]
4. Hence find an equation for each of the two tangents to the ellipse \(9x^2 + 16y^2 + 18x - 64y = c\) that are parallel to the line \(y = x\). [3 marks]

In this question you must show detailed reasoning. The function f is defined by \(\text{f}(x) = \cos x + \sqrt{3} \sin x\) with domain \(0 \leqslant x \leqslant 2\pi\).

1. Solve the following equations.
   1. \(\text{f}'(x) = 0\) [4]
   2. \(\text{f}''(x) = 0\) [3]
   The diagram shows the graph of the gradient function \(y = \text{f}'(x)\) for the domain \(0 \leqslant x \leqslant 2\pi\). \includegraphics{figure_5}
2. Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points \(A\), \(B\), \(C\) and \(D\). [2]
3. 1. Explain how to use the graph of the gradient function to find the values of \(x\) for which f(x) is increasing. [1]
   2. Using set notation, write down the set of values of \(x\) for which f(x) is increasing in the domain \(0 \leqslant x \leqslant 2\pi\). [2]

\includegraphics{figure_1} The diagram shows triangle \(ABC\), with \(AC = 13.5\) cm, \(BC = 8.3\) cm and angle \(ABC = 32°\). Find angle \(CAB\). [2]

The points \(P\) and \(Q\) have coordinates \((2, -5)\) and \((3, 1)\) respectively. Determine the equation of the circle that has \(PQ\) as a diameter. Give your answer in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

A circle \(C\) has equation \(x^2 + y^2 - 6x + 10y + k = 0\).

1. Find the set of possible values of \(k\). [2]
2. It is given that \(k = -46\). Determine the coordinates of the two points on \(C\) at which the gradient of the tangent is \(\frac{1}{2}\). [5]

\(AB\) is a diameter of a circle where \(A\) is \((1, 4)\) and \(B\) is \((7, -2)\)

1. Find the coordinates of the midpoint of \(AB\). [1 mark]
2. Show that the equation of the circle may be written as $$x^2 + y^2 - 8x - 2y = 1$$ [4 marks]
3. The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\). Find the exact area of triangle \(DEC\). [4 marks]

A circle of radius 6 passes through the points \((0, 0)\) and \((0, 10)\).

1. Sketch the two possible positions of the circle. [1 mark]
2. Find the equations of the two circles. [3 marks]

A circle has equation $$x^2 + y^2 + 10x - 4y - 71 = 0$$

1. Find the centre of the circle. [2 marks]
2. Hence, find the equation of the tangent to the circle at the point \((1, 10)\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [4 marks]

The circle with equation \((x - 7)^2 + (y + 2)^2 = 5\) has centre C.

1. 1. Write down the radius of the circle. [1 mark]
   2. Write down the coordinates of C. [1 mark]
2. The point \(P(5, -1)\) lies on the circle. Find the equation of the tangent to the circle at \(P\), giving your answer in the form \(y = mx + c\) [4 marks]
3. The point Q(3, 3) lies outside the circle and the point T lies on the circle such that QT is a tangent to the circle. Find the length of QT. [4 marks]

A circle with centre \(C\) has equation \(x^2 + y^2 + 8x - 12y = 12\)

1. Find the coordinates of \(C\) and the radius of the circle. [3 marks]
2. The points \(P\) and \(Q\) lie on the circle. The origin is the midpoint of the chord \(PQ\). Show that \(PQ\) has length \(n\sqrt{3}\), where \(n\) is an integer. [5 marks]

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]

A circle has equation \(x^2 + y^2 - 6x - 8y = 264\) \(AB\) is a chord of the circle. The angle at the centre of the circle, subtended by \(AB\), is \(0.9\) radians, as shown in the diagram below. \includegraphics{figure_5} Find the area of the minor segment shaded on the diagram. Give your answer to three significant figures. [5 marks]