Geo5.1.48

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P is a point on the line lx+my+n=0\,.The polar of P w.r.t the parabola y^{2}=4ax\, meets the curve in Q and R.Show that the locus of the midpoint of QR is l(y^{2}-4ax)+2a(lx+my+n)=0\,

Let M(x1,y1) be the midpoint of QR

Therefore,equation to the line QR is yy_{1}-2a(x+x_{1})=y_{1}^{{2}}-4ax_{1}\,

2ax-yy_{1}+y_{1}^{{2}}-2ax_{1}=0\,

Pole of this equation is

P\left({\frac  {y_{1}^{{2}}-2ax_{1}}{2a}},y_{1}\right)\,

The point lies on the line lx+my+n=0\,

Locus of M is l(y^{2}-2ax)+2a(my+n)=0\,

l(y^{2}-4ax)+2a(lx+my+n)=0\,


Main Page:Geometry:The Parabola