Parabola
14.0 Chord of contact
14.0 Chord of contact
The chord joining the points of contact of two tangents drawn from an external point to a parabola is called its chord of contact. The equation of chord of contact can be find out using $T=0$ i.e., $$y{y_1} = 2a\left( {x + {x_1}} \right)$$
Proof: Let $PQ$ 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 $Q({x'},{y'})$ and $R({x^{''}},{y^{''}})$ is the chord of contact $QR$.
Equation of tangent $PQ$ using $T=0$ is $$yy' = 2a\left( {x + x'} \right)...(1)$$
Equation of tangent $PR$ using $T=0$ is $$yy'' = 2a\left( {x + x''} \right)...(2)$$
Since lines given in equation $(1)$ and $(2)$ both pass through a point $P({x_1},{y_1})$, then $${y_1}y' = 2a\left( {{x_1} + x'} \right)$$ and $${y_1}y'' = 2a\left( {{x_1} + x''} \right)$$
Therefore, points $P({x_1},{y_1})$ and $Q({x'},{y'})$ lie on $y{y_1} = 2a\left( {x + {x_1}} \right)$ is the equation of chord of contact.
