Let \(x = 9\) be a directrix of an ellipse centred at \((0, 0)\) and having eccentricity \(\frac{1}{3}\). If focus at \((\alpha, 0)\) (\(\alpha<0\)), then locus of the mid-point of the chord passing through the focus \((\alpha, 0)\) is
Show Hint
The equation of a chord of any conic with mid-point \((x_1, y_1)\) is always \(T = S_1\). This is a very efficient shortcut for locus problems involving mid-points.