A circle with centre C has equation x^2+8x+y^2-12y=12. The points P and Q lie on the circle. The origin is the midpoint of the chord PQ. Show that PQ has length nsqrt(3) , where n is an integer.

First complete the square for both x and y. Move all constants to the right hand side. The square root of this is the radius of the circle. The two constants in the completed square bracket show the x and y coordinate of the centre of the circle.
Now, using this information you know that both P and Q are the radius away from the centre. Work out the distance from the centre to the origin point. You now have two sides of a triangle (much easier to show with diagram and how much detail this part would need to be gone into depends on the level of the student). Use Pythagoras to find the distance PO and double this to find distance PQ.