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

The circle \(C\) has radius 5 and its centre is the origin. The point \(T\) has coordinates \((11, 0)\). The tangents from \(T\) to the circle \(C\) touch \(C\) at the points \(R\) and \(S\).

1. Write down the geometrical name for the quadrilateral \(ORTS\). [1]
2. Find the exact value of the area of the quadrilateral \(ORTS\). Give your answer in its simplest form. [5]

\includegraphics{figure_7} The diagram shows a circle which passes through the points \(A(2, 9)\) and \(B(10, 3)\). \(AB\) is a diameter of the circle.

1. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the exact coordinates of \(C\). [4]
2. Find the exact area of the triangle formed by \(B\), \(C\) and the centre of the circle [3]

The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).

1. Find the centre and radius of the circle. [3]
2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]

The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).

1. Find the centre and radius of the circle. [3]
2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]

In this question you must show detailed reasoning. A circle touches the lines \(y = \frac{1}{2}x\) and \(y = 2x\) at \((6, 3)\) and \((3, 6)\) respectively. \includegraphics{figure_6} Find the equation of the circle. [7]

In this question you must show detailed reasoning. The centre of a circle C is the point (-1, 3) and C passes through the point (1, -1). The straight line L passes through the points (1, 9) and (4, 3). Show that L is a tangent to C. [7]

A circle has centre \(C(3, -8)\) and radius \(10\).

1. Express the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = k$$ [2 marks]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. [3 marks]
3. The line with equation \(y = 2x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x^2 + 6x - 2 = 0$$ [3 marks]
   2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt{n}\), where \(m\) and \(n\) are integers. [2 marks]

A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5x^2 + (2r - 48)x + (r^2 - 24r + 144) = 0$$ [3]

Given also that \(l\) is a tangent to \(C\),

1. find the two possible values of \(r\), giving your answers as fully simplified surds. [4]

\includegraphics{figure_4} A circle with centre \((9, -6)\) touches the \(x\)-axis as shown in Figure 4.

1. Write down an equation for the circle. [3] A line \(l\) is parallel to the \(x\)-axis. The line \(l\) cuts the circle at points \(P\) and \(Q\). Given that the distance \(PQ\) is 8
2. find the two possible equations for \(l\). [4]

$$x^2 + y^2 - 2x - 2y = 8$$ The circle with the above equation has radius \(r\) and has its centre at the point \(C\).

1. Determine the value of \(r\) and the coordinates of \(C\). [3]

The point \(P(4,2)\) lies on the circle.

1. Show that an equation of the tangent to the circle at \(P\) is [4] $$y = 14 - 3x.$$

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]

A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.

1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
2. 1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
   2. Give a geometrical interpretation of the case in part (b)(i). [1]
3. Give a geometrical interpretation of the case where \(a = 5\). [1]

A circle, C, has equation \(x^2 - 6x + y^2 = 16\). A second circle, D, has the following properties:

• The line through the centres of circle C and circle D has gradient 1.
• Circle D touches circle C at exactly one point.
• The centre of circle D lies in the first quadrant.
• Circle D has the same radius as circle C.

Find the coordinates of the centre of circle D. [5]

A line has equation \(y = 2x\) and a circle has equation \(x^2 + y^2 + 2x - 16y + 56 = 0\).

1. Show that the line does not meet the circle. [3]
2. 1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2x\). [4]
   2. Hence find the shortest distance between the line \(y = 2x\) and the circle, giving your answer in an exact form. [4]

The circle \(x^2 + y^2 + 2x - 14y + 25 = 0\) has its centre at the point C. The line \(7y = x + 25\) intersects the circle at points A and B. Prove that triangle ABC is a right-angled triangle. [7]

The circle \(C\) has equation $$x^2 + y^2 + 10x - 4y + 1 = 0$$

1. Find
   1. the coordinates of the centre of \(C\)
   2. the exact radius of \(C\) [2]
   The line with equation \(y = k\), where \(k\) is a constant, cuts \(C\) at two distinct points.
2. Find the range of values for \(k\), giving your answer in set notation. [2]
3. The line with equation \(y = mx + 4\) is a tangent to \(C\). Find possible exact values of \(m\). [5]

The circles \(C_1\) and \(C_2\) have respective equations $$x^2 + y^2 - 6x - 2y = 15$$ $$x^2 + y^2 - 18x + 14y = 95.$$

1. By considering the coordinates of the centres and the lengths of the radii of \(C_1\) and \(C_2\), show that \(C_1\) and \(C_2\) touch internally at some point \(P\). [4]
2. Determine the coordinates of \(P\). [3]
3. Find the equation of the common tangent to the circles at P. [4]

\includegraphics{figure_3} The diagram shows a circle with centre \((a, -a)\) that passes through the origin.

1. Write down an equation for the circle in terms of \(a\). [2]
2. Given that the point \((1, -5)\) lies on the circle, find the exact area of the circle. [3]

The diagram shows the circle with centre O and radius 2, and the parabola \(y = \frac{1}{\sqrt{3}}(4 - x^2)\). \includegraphics{figure_5} The circle meets the parabola at points \(P\) and \(Q\), as shown in the diagram.

1. Verify that the coordinates of \(Q\) are \((1, \sqrt{3})\). [3]
2. Find the exact area of the shaded region enclosed by the arc \(PQ\) of the circle and the parabola. [8]

The point \(F\) has coordinates \((0, a)\) and the straight line \(D\) has equation \(y = b\), where \(a\) and \(b\) are constants with \(a > b\). The curve \(C\) consists of points equidistant from \(F\) and \(D\).

1. Show that the cartesian equation of \(C\) can be expressed in the form $$y = \frac{1}{2(a-b)}x^2 + \frac{1}{2}(a+b).$$ [3]
2. State the \(y\)-coordinate of the lowest point of the curve and prove that \(F\) and \(D\) are on opposite sides of \(C\). [2]
3. 1. The point \(P\) on the curve has \(x\)-coordinate \(\sqrt{a^2 - b^2}\), where \(|a| > |b|\). Show that the tangent at \(P\) passes through the origin. [4]
   2. The tangent at \(P\) intersects the line \(D\) at the point \(Q\). In the case that \(a = 12\) and \(b = -8\), find the coordinates of \(P\) and \(Q\). Show that the length of \(PQ\) can be expressed as \(p\sqrt{q}\), where \(p = 2q\). [5]

A circle has equation \(x^2 + y^2 = 16\). Find the volume generated when the region in the first quadrant which is bounded by the circle and the lines \(x = 1\) and \(x = 2\) is rotated through \(2\pi\) radians about the \(x\)-axis. [5]

\includegraphics{figure_2} The diagram shows a triangle \(ABC\). The vertices have coordinates \(A(3, -7)\), \(B(9, 1)\) and \(C(-1, -5)\).

1. 1. Find the length of the side \(AB\). [2]
   2. Find the coordinates of the mid-point of \(AB\). [1]
   3. A circle has diameter \(AB\). Find the equation of the circle in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\), \(b\) and \(r\) are constants to be found. [3]
2. Find the equation of the line \(l\) passing through \(B\) parallel to \(AC\). [3]

A circle, of radius \(\sqrt{5}\) and centre the origin \(O\), is divided into two segments by the line \(y = 1\).

1. Determine the area of the smaller segment. [4]

The line is rotated clockwise about \(O\) through \(45^{\circ}\) and then reflected in the \(x\)-axis.

1. Find the equation of the line in its final position. [3]