If the tangents of a parabola at the points (x', y') and (x", y") meet at the point $(x_{1}, y_{1})$ and the normals at the same points in $(x_{2}, y_{2})$ , Prove that :
$x_{1} = \frac{y^{'} y^{''}}{4 a}$ and $y_{1} = \frac{y^{'} + y "}{2},$

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Solution

If tangent of a parabola at point $(x^{'}, y^{'})$ and $(x ", y ")$ meet at point $(x_{1}, y_{1})$ and normals at some point in $(x_{2}, y_{2})$ Then $x_{1}$ and $y_{1}$ are :
Let
$(x^{'}, y^{'}) = (a t_{1}^{2}, 2 a t_{1}) (x ", y ") = (a t_{2}^{2}, 2 a t_{2}) (x_{1}, y_{1}) = (a t_{1} t_{2}, a (t_{1} + t_{2})) \frac{y^{'} y "}{4 a} = \frac{(2 a t_{1}) (2 a t_{2})}{4 a} = a t_{1} t_{2} \frac{y^{'} + y "}{2} = \frac{2 a t_{1} + 2 a t_{2}}{2} = a (t_{1} + t_{2}) = y_{1} x_{1} = \frac{y^{'} y "}{4 a} y_{1} = \frac{y^{'} + y "}{2}$

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