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

Circles \(C_1\) and \(C_2\) have equations $$x^2 + y^2 + 6x - 10y + 18 = 0 \text{ and } (x-9)^2 + (y+4)^2 - 64 = 0$$ respectively.

1. Find the distance between the centres of the circles. [4] \(P\) and \(Q\) are points on \(C_1\) and \(C_2\) respectively. The distance between \(P\) and \(Q\) is denoted by \(d\).
2. Find the greatest and least possible values of \(d\). [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]

Points \(A\) and \(B\) have coordinates \((4, 3)\) and \((8, -5)\) respectively. A circle with radius 10 passes through the points \(A\) and \(B\).

1. Show that the centre of the circle lies on the line \(y = \frac{1}{2}x - 4\). [4]
2. Find the two possible equations of the circle. [5]

\includegraphics{figure_3} The points \(A(-3, -2)\) and \(B(8, 4)\) are at the ends of a diameter of the circle shown in Fig. 3.

1. Find the coordinates of the centre of the circle. [2]
2. Find an equation of the diameter \(AB\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
3. Find an equation of tangent to the circle at \(B\). [3]

The line \(l\) passes through \(A\) and the origin.

1. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions. [4]

The circle \(C\) has centre \(X(3, 5)\) and radius \(r\) The line \(l\) has equation \(y = 2x + k\), where \(k\) is a constant.

1. Show that \(l\) and \(C\) intersect when $$5x^2 + (4k - 26)x + k^2 - 10k + 34 - r^2 = 0$$ [3]

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

1. show that \(5r^2 = (k + p)^2\), where \(p\) is a constant to be found. [3]

\includegraphics{figure_2} The line \(l\)

• cuts the \(y\)-axis at the point \(A\)
• touches the circle \(C\) at the point \(B\)

as shown in Figure 2. Given that \(AB = 2r\)

1. find the value of \(k\) [6]

The points \(A\) and \(B\) have coordinates \((5, -1)\) and \((13, 11)\) respectively.

1. Find the coordinates of the mid-point of \(AB\). [2]

Given that \(AB\) is a diameter of the circle \(C\),

1. find an equation for \(C\). [4]

The circle \(C\), with centre at the point \(A\), has equation \(x^2 + y^2 - 10x + 9 = 0\). Find

1. the coordinates of \(A\), [2]
2. the radius of \(C\), [2]
3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. [2]

Given that the line \(l\) with gradient \(\frac{7}{T}\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),

1. find an equation of the line which passes through \(A\) and \(T\). [3]

\includegraphics{figure_1} In Figure 1, \(A(4, 0)\) and \(B(3, 5)\) are the end points of a diameter of the circle \(C\). Find

1. the exact length of \(AB\), [2]
2. the coordinates of the midpoint \(P\) of \(AB\), [2]
3. an equation for the circle \(C\). [3]

\includegraphics{figure_3} The points \(A\) and \(B\) lie on a circle with centre \(P\), as shown in Figure 3. The point \(A\) has coordinates \((1, -2)\) and the mid-point \(M\) of \(AB\) has coordinates \((3, 1)\). The line \(l\) passes through the points \(M\) and \(P\).

1. Find an equation for \(l\). [4]

Given that the \(x\)-coordinate of \(P\) is 6,

1. use your answer to part (a) to show that the \(y\)-coordinate of \(P\) is \(-1\). [1]
2. find an equation for the circle. [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]

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]

\includegraphics{figure_12} 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 circle \(C\) has equation \(x^2 + y^2 - 8x - 16y - 209 = 0\).

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

The point \(P(x, y)\) lies on \(C\).

1. Find, in terms of \(x\) and \(y\), the gradient of the tangent to \(C\) at \(P\). [3]
2. Hence or otherwise, find an equation of the tangent to \(C\) at the point \((21, 8)\). [2]

\includegraphics{figure_1} The points \(A(-3, -2)\) and \(B(8, 4)\) are at the ends of a diameter of the circle shown in Fig. 1.

1. Find the coordinates of the centre of the circle. [2]
2. Find an equation of the diameter \(AB\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
3. Find an equation of tangent to the circle at \(B\). [3]

The line \(l\) passes through \(A\) and the origin.

1. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions. [4]

\includegraphics{figure_1} The points \(A\) and \(B\) have coordinates \((2, -3)\) and \((8, 5)\) respectively, and \(AB\) is a chord of a circle with centre \(C\), as shown in Fig. 1.

1. Find the gradient of \(AB\). [2]

The point \(M\) is the mid-point of \(AB\).

1. Find an equation for the line through \(C\) and \(M\). [5]

Given that the \(x\)-coordinate of \(C\) is 4,

1. find the \(y\)-coordinate of \(C\), [2]
2. show that the radius of the circle is \(\frac{5\sqrt{17}}{4}\). [4]

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]

A circle \(C\) has equation \(x^2 + y^2 + 8y - 24 = 0\).

1. Find the centre and radius of the circle. [3]
2. The point \(A(2, 2)\) lies on the circumference of \(C\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]

\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]

A circle has equation \((x - 5)^2 + (y - 2)^2 = 20\).

1. State the coordinates of the centre and the radius of this circle. [2]
2. State, with a reason, whether or not this circle intersects the \(y\)-axis. [2]
3. Find the equation of the line parallel to the line \(y = 2x\) that passes through the centre of the circle. [2]
4. Show that the line \(y = 2x + 2\) is a tangent to the circle. State the coordinates of the point of contact. [5]

\includegraphics{figure_11} 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]

\includegraphics{figure_13} Fig. 13 shows the circle with equation \((x - 4)^2 + (y - 2)^2 = 16\).

1. Write down the radius of the circle and the coordinates of its centre. [2]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. Give your answers in surd form. [4]
3. Show that the point A \((4 + 2\sqrt{2}, 2 + 2\sqrt{2})\) lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it. Find the equations of both these tangents. [7]

The circle \((x - 3)^2 + (y - 2)^2 = 20\) has centre C.

1. Write down the radius of the circle and the coordinates of C. [2]
2. Find the coordinates of the intersections of the circle with the \(x\)- and \(y\)-axes. [5]
3. Show that the points A\((1, 6)\) and B\((7, 4)\) lie on the circle. Find the coordinates of the midpoint of AB. Find also the distance of the chord AB from the centre of the circle. [5]

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]