Geo5.1.47

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From the points of 2x-3y+4=0\, tangents are drawn to y^{2}=4ax\,. Show that the chords of contact pass through a fixed point.

Let P be a point on the line 2x-3y+4=0\,

Let lx+my+n=0\, be the chord of contact of P w.r.t the parabola,then P=[{\frac  {n}{l}},{\frac  {-2am}{l}}]\,

P lies on the given line, then 2({\frac  {n}{l}})-3({\frac  {-2am}{l}})+4=0\,

2l+3am+n=0\,

l(2)+m(3a)+n=0\,

That is, (2,3a)\, lies on lx+my+n=0\,

Hence the chord of contact of P passes through a fixed point (2,3a)


Main Page:Geometry:The Parabola