Geo5.1.45

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The line lx+my+na=0\, meets the parabola y^{2}=4ax\, at P,Q. The lines joining P and Q to the focus meet the parabola in M,N.Show that the equation to MN is nx-my+la=0\,

Let P(t_{1}),Q(t_{2})\, be the given points.

Therefore equation to PQ is y(t_{1}+t_{2})-2x-2at_{1}t_{2}=0\,

Given equation of PQ is lx+my+n=0\,. Let this be 2.

{\frac  {t_{1}+t_{2}}{m}}={\frac  {-2}{t}}={\frac  {-2at_{1}t_{2}}{na}}\,

t_{1}+t_{2}={\frac  {-2m}{l}},t_{1}t_{2}={\frac  {n}{l}}\,

Given PM,QN are the focal chords.If M and N are the points t3 and t4, then

t_{1}t_{3}=-1,t_{2}t_{4}=-1\,

Equation to MN is y(t_{3}+t_{4})-2x-2at_{3}t_{4}=0\,

y[{\frac  {-1}{t_{1}}}-{\frac  {-1}{t_{2}}}]-2x-{\frac  {2a}{t_{1}t_{2}}}=0\,

y(t_{1}+t_{2})+2xt_{1}t_{2}+2a=0\,

Substituting from 2,

y({\frac  {-2m}{l}})+2x{\frac  {n}{l}}+2a=0\,

Therefore,

nx-my+la=0\,


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