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in Co-ordinate geometry by (20 points)
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The locus of midpoints of all the chords of parabola y= 10x which passes through (-1,2) is

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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 \((y - 1)^2 = 5\left( x + \frac 65\right)\)

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