Solutions to this question by accurate drawing will not be accepted. A circle has centre \((4,2)\) and meets the \(x\)-axis at \((-2,0)\).

(a) Find the equation of the circle.

(b) Find, in exact form, the coordinates of the points where the circle meets the \(y\)-axis.

The point \(A\) has coordinates \((-2,4)\). The point \(B\) has coordinates \((6,10)\). The point \(C\) has coordinates \((12,2)\).

(a) Find the gradients of the lines \(AB\), \(AC\) and \(BC\).

(b) Hence find the equation of the circle which passes through the points \(A\), \(B\) and \(C\).

Point \(A\) has coordinates \((3,-1)\).

A circle has equation \((x-4)^2+(y+3)^2=5\).

(a) Show that \(A\) lies on the circumference of the circle.

(b) Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\).

(c) Find the equation of the tangent to the circle at \(A\).

A circle with centre \(C\) has the equation \(x^2+y^2-10x-4y+24=0\).

(a) Show that the line \(y=2x-3\) is a tangent to this circle.

(b) Given that this tangent touches the circle at the point \(P\), find the coordinates of \(P\).

(c) Find the equation of the circle which has its centre at \(P\) and passes through the origin.

Solutions by accurate drawing will not be accepted. A circle, \(C\), has equation \((x-5)^{2}+(y-12)^{2}=100\). (a) Find the equation of the tangent to \(C\) at the point \((11,4)\).

Give your answer in the form \(a x+b y=c\), where \(a, b\) and \(c\) are integers. (b) Show that \(C\) and the circle with equation \(x^{2}+y^{2}=4\) do not intersect.

The point \(A\) has coordinates \((-3,6)\).

The point \(B\) has coordinates \((7,-8)\).

Given that the line \(AB\) is the diameter of a circle, find the equation of the circle.