A circle has equation $(x - 1)^{2} + (y + 2)^{2} = 25$ . Find the equation of the tangent to the circle at the point $P (4, 2)$ .

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Find the equation of the line passing through $(3, - 2)$ perpendicular to the line with equation $2 x - 4 y + 7 = 0$ .
Find the centre and radius of the circle with equation $x^{2} + y^{2} - 6 x + 8 y - 11 = 0$ .
Triangle $A B C$ has vertices $A (- 2, 1), B (4, 3)$ , and $C (2, 7)$ . Find the equation of the median from vertex $C$ .
Find the range of values of $c$ for which the line $y = 2 x + c$ intersects the circle $x^{2} + y^{2} = 5$ at two distinct points.
Determine the distance between the parallel lines $3 x + 4 y = 10$ and $3 x + 4 y = - 15$ .