6 A circle has centre \(C ( - 3,1 )\) and radius \(\sqrt { 13 }\).
- Express the equation of the circle in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
- Hence find the equation of the circle in the form
$$x ^ { 2 } + y ^ { 2 } + m x + n y + p = 0$$
where \(m , n\) and \(p\) are integers.
- The circle cuts the \(y\)-axis at the points \(A\) and \(B\). Find the distance \(A B\).
- Verify that the point \(D ( - 5 , - 2 )\) lies on the circle.
- Find the gradient of \(C D\).
- Hence find an equation of the tangent to the circle at the point \(D\).