Normal to circle at point

Find the equation of the normal (line through centre and point) to a circle at a given point.

4 questions · Moderate -0.4

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Find
1. the coordinates of the centre of $C$,
2. the exact radius of $C$. The point $P$ lies on $C$.
  Given that the tangent to $C$ at $P$ has equation $x + 2 y + 10 = 0$
find the coordinates of $P$
Find the equation of the normal to $C$ at $P$, giving your answer in the form $y = m x + c$ where $m$ and $c$ are integers to be found.

1. Show that the $x$-coordinates of the points $A$ and $C$ satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
2. Find the coordinates of the points $A$ and $C$.
Find $\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x$.
Find the area of the shaded region $R$ bounded by the curve and the line segment $A B$.
[0pt] [4 marks] $7 \quad$ A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0$. The point $A ( 3 , - 2 )$ lies on the circle.

Express the equation in the form $(x - a)^2 + (y - b)^2 = r^2$, where $a$ is to be given in terms of $p$ and $r^2$ is to be given in terms of $p$ and $q$. [2]

1. Find the equation of the normal to the circle at the point $A$. [3]
2. Find the values of $p$ and $q$. [5]