A circle has equation \(x^{2}+y^{2}-25=0\).
A second circle has the same radius as the first circle, and the coordinates of its centre are both positive.
The two circles intersect at the points \(A\) and \(B\).
The line \(A B\) has length 6 and is parallel to the line \(y=-x\).
Find the equation of the second circle in the form \(x^{2}+y^{2}+a x+b y+c=0\), where \(a, b\) and \(c\) are constants.

The lines \(x=0\), \(x=4\), \(y=3\) and \(y=-1\) are tangents to a circle.

(a) Find the equation of the circle.

The line \(y=2x+a\), where \(a\) is a constant, is also a tangent to the circle.

(b) Show that \(5x^2+4(a-2)x+(a-1)^2=0\), and hence find the possible values of \(a\). Give your answers in exact form.

The line with equation

\(x+3y=k,\)

where \(k\) is a positive constant, is a tangent to the curve with equation

\(x^2+y^2+2y-9=0.\)

Find the value of \(k\) and hence find the coordinates of the point where the line touches the curve.

The curves

\(y=x^2 \quad\text{and}\quad y^2=27x\)

intersect at \(O(0,0)\) and at the point \(A\). Find the equation of the perpendicular bisector of the line \(OA\).

Find the coordinates of the points where the line \(2y-3x=6\) intersects the curve

\(\frac{x^2}{4}+\frac{y^2}{9}=5.\)

The line \(y=kx+3\), where \(k\) is a positive constant, is a tangent to the curve \(x^2-2x+y^2=8\) at the point \(P\).

(i) Find the value of \(k\).

(ii) Find the coordinates of \(P\).

(iii) Find the equation of the normal to the curve at \(P\).