# Geo5.1.49

Show that the tangent at one extremity of a focal chord of a parabola is parallel to the normal at the other extremity.

Let $P(at_{1}^{{2}},2at_{1}),Q(at_{2}^{{2}},2at_{2})\,$ be the two extremeties of a focal chord

of the given parabola.

$t_{1}t_{2}=-1\,$

Equation of the tangent at P is $yt_{1}-x-at_{1}^{{2}}=0\,$. equation 2

Slope of this equation is ${\frac {1}{t_{1}}}\,$.

Equation of the normal at Q is $y+t_{2}x=2at_{2}+at_{2}^{{3}}\,$

Slope of the normal is $-t_{2}=-{\frac {-1}{t_{1}}}={\frac {1}{t_{1}}}\,$ which is the slope of

the equation 2.

Therefore tangent at P is parallel to the normal at Q.