14.1 Length of chord of contact (QR)

  Parabola

Let the equation of parabola be ${y^2} = 4ax$. $PQ$ and $PR$ be the tangents to the parabola ${y^2} = 4ax$ drawn from an external point $P({x_1},{y_1})$ and the line joining the points $Q(a{t_1}^2,2a{t_1})$ and $R(a{t_2}^2,2a{t_2})$ is the chord of contact $QR$ as shown in figure $30$. The coordinates of point of intersection of tangents is $(a{t_1}{t_2},a({t_1} + {t_2}))$ i.e., $${x_1} = a{t_1}{t_2}...(1)$$ and $${y_1} = a({t_1} + {t_2})...(2)$$

From equations $(1)$ and $(2)$, we can write it as $${t_1}{t_2} = \frac{{{x_1}}}{a}...(3)$$ and $${t_1} + {t_2} = \frac{{{y_1}}}{a}...(4)$$

Now, using distance formulae between points $Q$ and $R$, $$QR = \sqrt {{{\left( {a{t_1}^2 - a{t_2}^2} \right)}^2} + {{\left( {2a{t_1} - 2a{t_2}} \right)}^2}} $$ $$ = \sqrt {{a^2}{{\left( {{t_1} - {t_2}} \right)}^2}\left[ {{{\left( {{t_1} + {t_2}} \right)}^2} + 4} \right]} $$ $$ = \left| a \right|\left| {{t_1} - {t_2}} \right|\sqrt {{{\left( {{t_1} + {t_2}} \right)}^2} + 4} $$ $$ = \left| a \right|\sqrt {{{\left( {{t_1} + {t_2}} \right)}^2} - 4{t_1}{t_2}} \sqrt {{{\left( {{t_1} + {t_2}} \right)}^2} + 4} $$

From equations $(3)$ and $(4)$, $$ = \left| a \right|\sqrt {{{\left( {\frac{{{y_1}}}{a}} \right)}^2} - 4\frac{{{x_1}}}{a}} \sqrt {{{\left( {\frac{{{y_1}}}{a}} \right)}^2} + 4} $$ $$ = \frac{1}{{\left| a \right|}}\sqrt {({y_1}^2 - 4a{x_1})\left( {({y_1}^2 + 4{a^2}} \right)} $$