1.03f Circle properties: angles, chords, tangents

7 The circle \(S\) has centre \(C ( 8,13 )\) and touches the \(x\)-axis, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-4_444_755_356_641}

1. Write down an equation for \(S\), giving your answer in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. The point \(P\) with coordinates \(( 3,1 )\) lies on the circle.
   1. Find the gradient of the straight line passing through \(P\) and \(C\).
   2. Hence find an equation of the tangent to the circle \(S\) at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   3. The point \(Q\) also lies on the circle \(S\), and the length of \(P Q\) is 10 . Calculate the shortest distance from \(C\) to the chord \(P Q\).

6 A circle has centre \(C ( 10,4 )\). The \(x\)-axis is a tangent to the circle, as shown in Fig. 6. \begin{figure}[h] \end{figure}

1. Find the equation of the circle.
2. Show that the line \(y = x\) is not a tangent to the circle.
3. Write down the position vector of the midpoint of OC.

The coordinates of points \(A\), \(B\) and \(C\) are \((6, 4)\), \((p, 7)\) and \((14, 18)\) respectively, where \(p\) is a constant. The line \(AB\) is perpendicular to the line \(BC\).

1. Given that \(p < 10\), find the value of \(p\). [4]

A circle passes through the points \(A\), \(B\) and \(C\).

1. Find the equation of the circle. [3]
2. Find the equation of the tangent to the circle at \(C\), giving the answer in the form \(dx + ey + f = 0\), where \(d\), \(e\) and \(f\) are integers. [3]

The equation of a circle is \(x^2 + y^2 + px + 2y + q = 0\), where \(p\) and \(q\) are constants.

1. Express the equation in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\) is to be given in terms of \(p\) and \(r^2\) is to be given in terms of \(p\) and \(q\). [2]

The line with equation \(x + 2y = 10\) is the tangent to the circle at the point \(A(4, 3)\).

1. 1. Find the equation of the normal to the circle at the point \(A\). [3]
   2. Find the values of \(p\) and \(q\). [5]

\(A(-1, 1)\) and \(P(a, b)\) are two points, where \(a\) and \(b\) are constants. The gradient of \(AP\) is 2.

1. Find an expression for \(b\) in terms of \(a\). [2]
2. \(B(10, -1)\) is a third point such that \(AP = AB\). Calculate the coordinates of the possible positions of \(P\). [6]

The coordinates of two points \(A\) and \(B\) are \((1, 3)\) and \((9, -1)\) respectively and \(D\) is the mid-point of \(AB\). A point \(C\) has coordinates \((x, y)\), where \(x\) and \(y\) are variables.

1. State the coordinates of \(D\). [1]
2. It is given that \(CD^2 = 20\). Write down an equation relating \(x\) and \(y\). [1]
3. It is given that \(AC\) and \(BC\) are equal in length. Find an equation relating \(x\) and \(y\) and show that it can be simplified to \(y = 2x - 9\). [3]
4. Using the results from parts (ii) and (iii), and showing all necessary working, find the possible coordinates of \(C\). [4]

\includegraphics{figure_2} In the diagram, \(OADC\) is a sector of a circle with centre \(O\) and radius 3 cm. \(AB\) and \(CB\) are tangents to the circle and angle \(ABC = \frac{1}{4}\pi\) radians. Find, giving your answer in terms of \(\sqrt{3}\) and \(\pi\),

1. the perimeter of the shaded region, [3]
2. the area of the shaded region. [3]

\includegraphics{figure_1} The diagram shows a major arc \(AB\) of a circle with centre \(O\) and radius 6 cm. Points \(C\) and \(D\) on \(OA\) and \(OB\) respectively are such that the line \(AB\) is a tangent at \(E\) to the arc \(CED\) of a smaller circle also with centre \(O\). Angle \(COD = 1.8\) radians.

1. Show that the radius of the arc \(CED\) is 3.73 cm, correct to 3 significant figures. [2]
2. Find the area of the shaded region. [4]

The circle \(C\), with centre \(A\), has equation $$x^2 + y^2 - 6x + 4y - 12 = 0.$$

1. Find the coordinates of \(A\). [2]
2. Show that the radius of \(C\) is 5. [2]

The points \(P\), \(Q\) and \(R\) lie on \(C\). The length of \(PQ\) is 10 and the length of \(PR\) is 3.

1. Find the length of \(QR\), giving your answer to 1 decimal place. [3]

The hyperbola \(C\) has equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

1. Show that an equation of the normal to \(C\) at the point \(P(a \sec t, b \tan t)\) is $$ax \sin t + by = (a^2 + b^2) \tan t.$$ [6]

The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and \(S\) is a focus of \(C\). Given that the eccentricity of \(C\) is \(\frac{3}{2}\), and that \(OA = 3OS\), where \(O\) is the origin,

1. determine the possible values of \(t\), for \(0 \leq t < 2\pi\). [8]

A circle with centre \(C\) has equation \(x^2 + y^2 - 2x + 10y - 19 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle. [3]
2. Verify that the point \((7, -2)\) lies on the circumference of the circle. [1]
3. Find the equation of the tangent to the circle at the point \((7, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

The points \(A\) and \(B\) have coordinates \((4, -2)\) and \((10, 6)\) respectively. \(C\) is the mid-point of \(AB\). Find

1. the coordinates of \(C\), [2]
2. the length of \(AC\), [2]
3. the equation of the circle that has \(AB\) as a diameter, [3]
4. the equation of the tangent to the circle in part (iii) at the point \(A\), giving your answer in the form \(ax + by = c\). [5]

\includegraphics{figure_10} Fig. 10 shows a circle with centre C\((2, 1)\) and radius 5.

1. Show that the equation of the circle may be written as $$x^2 + y^2 - 4x - 2y - 20 = 0.$$ [3]
2. Find the coordinates of the points P and Q where the circle cuts the \(y\)-axis. Leave your answers in the form \(a \pm \sqrt{b}\). [3]
3. Verify that the point A\((5, -3)\) lies on the circle. Show that the tangent to the circle at A has equation \(4y = 3x - 27\). [6]

The points \(P\) and \(Q\) have coordinates \((-2, 6)\) and \((4, -1)\) respectively. Given that \(PQ\) is a diameter of circle \(C\),

1. find the coordinates of the centre of \(C\), [2]
2. show that \(C\) has the equation $$x^2 + y^2 - 2x - 5y - 14 = 0. \quad [5]$$

The point \(R\) has coordinates \((2, 7)\).

1. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle PRQ\) in degrees. [2]

The circle \(C\) has the equation $$x^2 + y^2 + 10x - 8y + k = 0,$$ where \(k\) is a constant. Given that the point with coordinates \((-6, 5)\) lies on \(C\),

1. find the value of \(k\), [2]
2. find the coordinates of the centre and the radius of \(C\). [3]

A straight line which passes through the point \(A(2, 3)\) is a tangent to \(C\) at the point \(B\).

1. Find the length \(AB\) in the form \(k\sqrt{5}\). [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]

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]

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]

\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]

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]

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]

In this question you must show detailed reasoning. The lines \(y = \frac{1}{2}x\) and \(y = -\frac{1}{2}x\) are tangents to a circle at \((2, 1)\) and \((-2, 1)\) respectively. Find the equation of the circle in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are constants. [6]

In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]