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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.


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.

Main Page:Geometry:The Parabola