Question 7 - A-Level Maths

7 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$.

Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
Write down:
1. the coordinates of $C$;
2. the radius of the circle.
The point $D$ has coordinates (7, -2).
1. Verify that the point $D$ lies on the circle.
2. Find an equation of the normal to the circle at the point $D$, giving your answer in the form $m x + n y = p$, where $m , n$ and $p$ are integers.
1. A line has equation $y = k x$. Show that the $x$-coordinates of any points of intersection of the line and the circle satisfy the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$
2. Find the values of $k$ for which the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$ has equal roots.
3. Describe the geometrical relationship between the line and the circle when $k$ takes either of the values found in part (d)(ii).