Geo5.1.50

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Prove that the length of the chord of contact of tangents drawn from (x_{1},y_{1})\, to the parabola y^{2}=4ax\, is {\frac  {{\sqrt  {(y_{1}^{{2}}-4ax_{1})}}{\sqrt  {(y_{1}^{{2}}+4a^{2})}}}{a}}\,.

Let Q(at_{1}^{{2}},2at_{1}),R(at_{2}^{{2}},2at_{2})\, be the two points of contact of tangents

from P(x_{1},y_{1})\, to the given parabola.

P=point of intersection of the tangents at Q and R is [at_{1}t_{2},a(t_{1}+t_{2})]\,

Therefore x_{1}=at_{1}t_{2},y_{1}=a(t_{1}+t_{2})\,

t_{1}t_{2}={\frac  {x_{1}}{a}},t_{1}+t_{2}={\frac  {y_{1}}{a}}\,

QR=length of the chord of contact is

QR={\sqrt  {a^{2}(t_{1}^{{2}}-t_{2}^{{2}})^{2}+4a^{2}(t_{1}-t_{2})^{2}}}={\sqrt  {a^{2}(t_{1}-t_{2})^{2}[(t_{1}+t_{2})^{2}+4]}}\,

{\frac  {{\sqrt  {(y_{1}^{{2}}-4ax_{1})}}{\sqrt  {(y_{1}^{{2}}+4a^{2})}}}{a}}\,.


Main Page:Geometry:The Parabola