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

A circle \(C\) has equation \(x^2 + y^2 - 6x + 10y + k = 0\).

1. Find the set of possible values of \(k\). [2]
2. It is given that \(k = -46\). Determine the coordinates of the two points on \(C\) at which the gradient of the tangent is \(\frac{1}{2}\). [5]

A circle has equation \((x - 2)^2 + (y + 3)^2 = 13\) Find the gradient of the tangent to this circle at the origin. Circle your answer. [1 mark] \(-\frac{3}{2}\) \quad \(-\frac{2}{3}\) \quad \(\frac{2}{3}\) \quad \(\frac{3}{2}\)

\(AB\) is a diameter of a circle where \(A\) is \((1, 4)\) and \(B\) is \((7, -2)\)

1. Find the coordinates of the midpoint of \(AB\). [1 mark]
2. Show that the equation of the circle may be written as $$x^2 + y^2 - 8x - 2y = 1$$ [4 marks]
3. The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\). Find the exact area of triangle \(DEC\). [4 marks]

A circle of radius 6 passes through the points \((0, 0)\) and \((0, 10)\).

1. Sketch the two possible positions of the circle. [1 mark]
2. Find the equations of the two circles. [3 marks]

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 line \(L_1\) has equation \(x + 7y - 41 = 0\) \(L_1\) is a tangent to the circle C at the point P(6, 5) The line \(L_2\) has equation \(y = x + 3\) \(L_2\) is a tangent to the circle C at the point Q(0, 3) The lines \(L_1\) and \(L_2\) and the circle C are shown in the diagram below. \includegraphics{figure_11}

1. Show that the equation of the radius of the circle through P is \(y = 7x - 37\) [3 marks]
2. Find the equation of C [4 marks]

Point \(A\) has coordinates \((4, 1)\) and point \(B\) has coordinates \((-8, 5)\)

1. Find the equation of the perpendicular bisector of \(AB\) [5 marks]
2. A circle passes through the points \(A\) and \(B\) A diameter of the circle lies along the \(x\)-axis. Find the equation of the circle. [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]

A circle with centre \(C\) has equation \(x^2 + y^2 + 8x - 12y = 12\)

1. Find the coordinates of \(C\) and the radius of the circle. [3 marks]
2. The points \(P\) and \(Q\) lie on the circle. The origin is the midpoint of the chord \(PQ\). Show that \(PQ\) has length \(n\sqrt{3}\), where \(n\) is an integer. [5 marks]

The line \(L\) has equation $$5y + 12x = 298$$ A circle, \(C\), has centre \((7, 9)\) \(L\) is a tangent to \(C\).

1. Find the coordinates of the point of intersection of \(L\) and \(C\). Fully justify your answer. [5 marks]
2. Find the equation of \(C\). [3 marks]

One of the equations below is the equation of a circle. Identify this equation. [1 mark] Tick \((\checkmark)\) one box. \((x + 1)^2 - (y + 2)^2 = -36\) \((x + 1)^2 - (y + 2)^2 = 36\) \((x + 1)^2 + (y + 2)^2 = -36\) \((x + 1)^2 + (y + 2)^2 = 36\)

A circle has equation \((x - 4)^2 + (y + 4)^2 = 9\) What is the area of the circle? Circle your answer. [1 mark] \(3\pi\) \quad \(9\pi\) \quad \(16\pi\) \quad \(81\pi\)

A circle has equation \(x^2 + y^2 - 6x - 8y = 264\) \(AB\) is a chord of the circle. The angle at the centre of the circle, subtended by \(AB\), is \(0.9\) radians, as shown in the diagram below. \includegraphics{figure_5} Find the area of the minor segment shaded on the diagram. Give your answer to three significant figures. [5 marks]