Show that the tangent to the curve \(y = (x+k)^2\) at the point where \(x = 2k\) is \[ y + 3k^2 = 6kx. \]
To find the gradient, it might help to expand the bracket.
This tangent meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). The mid-point of \(PQ\) is \(M\).
Find the co-ordinates of \(M\) in terms of \(k\) and hence deduce the equation of the locus of \(M\) as \(k\) varies.
Given two points in the plane whose coordinates are \((x_1,y_1)\) and \((x_2,y_2)\), can we write down the coordinates of their midpoint?