Find $a \cdot b$ for $a = - 4 i + 3 j - 4 k$ and $b = 4 i + 0 j + 3 k$ .

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The point $P (2, 3)$ lies outside the circle with centre $(- 1, - 1)$ and radius $3$ . Find the length of a tangent from $P$ to the circle.
A bag contains 5 red and 7 blue balls. Two balls are drawn without replacement. Find the probability that both are red.
Find the radius of the circle $x^{2} + y^{2} - 8 x - 10 y + 16 = 0$ .
Find the $y$ -intercept of the line through $(- 3, - 9)$ and $(- 1, - 7)$ .
In how many ways can $5$ different books be arranged in a row on a shelf if two particular books must be together?
Find the radius of the circle $x^{2} + y^{2} + 8 x + 8 y - 17 = 0$ .