Equation of chord having P(x1, y1) as mid-point is
\(T = S_1\)
⇒ \(yy_1 - 5 = (x + x_1) = {y_1}^2 - 10x_1\)
It passes through (-1, 2)
∴ \(2y_1 - 5 (-1 + x_1) = {y_1}^2 - 10x\)
∴ Locus of P is \(y^2 - 10x = 2y + 5 - 5x\)
⇒ \(y^2 - 5x - 2y - 5 = 0\)
⇒ \((y - 1)^2 - 5x - 6 = 0\)
⇒ \((y-1)^2 = 5x + 6\)
⇒ \((y - 1)^2 = 5\left( x + \frac 65\right)\)
Hence, locus of mid-point is \((y - 1)^2 = 5\left( x + \frac 65\right)\)