1.03d Circles: equation (x-a)^2+(y-b)^2=r^2

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]

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]

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 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]

Figure 1 \includegraphics{figure_1} The circle C, with centre (a, b) and radius 5, touches the x-axis at (4, 0), as shown in Fig. 1.

1. Write down the value of a and the value of b. [1]
2. Find a cartesian equation of C. [2]

A tangent to the circle, drawn from the point P(8, 17), touches the circle at T.

1. Find, to 3 significant figures, the length of PT. [3]

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 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]

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 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]

\includegraphics{figure_1} The circle \(C\), with centre \((a, b)\) and radius \(5\), touches the \(x\)-axis at \((4, 0)\), as shown in Fig. 1.

1. Write down the value of \(a\) and the value of \(b\). [1]
2. Find a cartesian equation of \(C\). [2]

A tangent to the circle, drawn from the point \(P(8, 17)\), touches the circle at \(T\).

1. Find, to 3 significant figures, the length of \(PT\). [3]

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]

Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below. \includegraphics{figure_5}

1. Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$ Write down the coordinates of Q in a similar form. [4] Now let the variable point P be defined by the parametric equations (*) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \to \infty\) and as \(a \to -\infty\). Show algebraically that this locus has an asymptote at \(y = -1\). On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies. [8] (The single curve made up of these two loci and including the point B is called a right strophoid.)
4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? [3]

\includegraphics{figure_1} The circle, with centre \(C\) and radius \(r\), touches the \(y\)-axis at \((0, 4)\) and also touches the line with equation \(4y - 3x = 0\), as shown in Fig. 1.

1. 1. Find the value of \(r\).
   2. Show that \(\arctan \left(\frac{4}{3}\right) + 2 \arctan \left(\frac{1}{2}\right) = \frac{1}{2} \pi\). [8]

The line with equation \(4x + 3y = q\), \(q > 12\), is a tangent to the circle.

1. Find the value of \(q\). [4]

\(A\) and \(B\) are fixed points in the \(x\)-\(y\) plane. The position vectors of \(A\) and \(B\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively. State, with reference to points \(A\) and \(B\), the geometrical significance of

1. the quantity \(|\mathbf{a} - \mathbf{b}|\), [1]
2. the vector \(\frac{1}{2}(\mathbf{a} + \mathbf{b})\). [1]

The circle \(P\) is the set of points with position vector \(\mathbf{p}\) in the \(x\)-\(y\) plane which satisfy $$\left|\mathbf{p} - \frac{1}{2}(\mathbf{a} + \mathbf{b})\right| = \frac{1}{2}|\mathbf{a} - \mathbf{b}|.$$

1. State, in terms of \(\mathbf{a}\) and \(\mathbf{b}\),
   1. the position vector of the centre of \(P\), [1]
   2. the radius of \(P\). [1]

It is now given that \(\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix}\).

1. Find a cartesian equation of \(P\). [4]

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]