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

5 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 5 . Find the coordinates of the centre of the circle and the value of \(k\).

1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).

1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
2. Find the coordinates of the midpoint of the chord \(A B\).

1 The equation of a circle is given by \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = r ^ { 2 }\).

1. Write down the coordinates of the centre of the circle.
2. The circle passes through the point \(( 0,2 )\). Find the length of the diameter.

1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).

1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
2. Find the coordinates of the midpoint of the chord \(A B\).

5 A circle \(S\) has centre at the point \(( 3,1 )\) and passes through the point \(( 0,5 )\).

1. Find the radius of \(S\) and hence write down its cartesian equation.
2. (a) Determine the two points on \(S\) where the \(y\)-coordinate is twice the \(x\)-coordinate.
   (b) Calculate the length of the minor arc joining these two points.

The coordinates of three points \(A , B , C\) are \(( 4,6 ) , ( - 3,5 )\) and \(( 5 , - 1 )\) respectively. a) Show that \(B \widehat { A C }\) is a right angle.
b) A circle passes through all three points \(A , B , C\). Determine the equation of the circle.

The equation of a circle is \((x - a)^2 + (y - 3)^2 = 20\). The line \(y = \frac{1}{2}x + 6\) is a tangent to the circle at the point \(P\).

1. Show that one possible value of \(a\) is 4 and find the other possible value. [5]
2. For \(a = 4\), find the equation of the normal to the circle at \(P\). [4]
3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b). [4]

The equation of a circle is \((x - 3)^2 + y^2 = 18\). The line with equation \(y = mx + c\) passes through the point \((0, -9)\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle. [8]

The equation of a circle is \((x-6)^2 + (y+a)^2 = 18\). The line with equation \(y = 2a - x\) is a tangent to the circle.

1. Find the two possible values of the constant \(a\). [5]
2. For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent. [3]

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]

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]

\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]

\includegraphics{figure_2} Figure 2 shows the curve with equation \(y^2 = 4(x - 2)\) and the line with equation \(2x - 3y = 12\). The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).

1. Write down the coordinates of \(A\). [1]
2. Find, using algebra, the coordinates of \(P\) and \(Q\). [6]
3. Show that \(\angle PAQ\) is a right angle. [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]

A circle \(C\) has centre \(M\) \((6, 4)\) and radius 3.

1. Write down the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = r^2.$$ [2]

\includegraphics{figure_3} Figure 3 shows the circle \(C\). The point \(T\) lies on the circle and the tangent at \(T\) passes through the point \(P\) \((12, 6)\). The line \(MP\) cuts the circle at \(Q\).

1. Show that the angle \(TMQ\) is 1.0766 radians to 4 decimal places. [4]

The shaded region \(TPQ\) is bounded by the straight lines \(TP\), \(QP\) and the arc \(TQ\), as shown in Figure 3.

1. Find the area of the shaded region \(TPQ\). Give your answer to 3 decimal places. [5]

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]

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]