JEE Main & Advanced Mathematics Circle and System of Circles Chord of Contact of Tangents

(1) Chord of contact : The chord joining the points of contact of the two tangents to a conic drawn from a given point, outside it, is called the chord of contact of tangents.

(2) Equation of chord of contact : The equation of the chord of contact of tangents drawn from a point \[({{x}_{1}},\,{{y}_{1}})\] to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[x{{x}_{1}}+y{{y}_{1}}={{a}^{2}}.\]

Equation of chord of contact at \[({{x}_{1}},\,{{y}_{1}})\] to the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] is \[x{{x}_{1}}+y{{y}_{1}}+g(x+{{x}_{1}})+f(y+{{y}_{1}})+c=0\].

It is clear from above that the equation to the chord of contact coincides with the equation of the tangent, if point \[({{x}_{1}},\,{{y}_{1}})\] lies on the circle.

The length of chord of contact \[=2\sqrt{{{r}^{2}}-{{p}^{2}}}\]; (p being length of perpendicular from centre to the chord)

Area of \[\Delta APQ\] is given by \[\frac{a{{(x_{1}^{2}+y_{1}^{2}-{{a}^{2}})}^{3/2}}}{x_{1}^{2}+y_{1}^{2}}\].

(3) Equation of the chord bisected at a given point : The equation of the chord of the circle \[S\equiv {{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] bisected at the point \[({{x}_{1}},\,{{y}_{1}})\] is given by  \[T={{S}_{1}}\].

i.e., \[x{{x}_{1}}+y{{y}_{1}}+g(x+{{x}_{1}})+f(y+{{y}_{1}})+c=x_{1}^{2}+y_{1}^{2}+2g{{x}_{1}}+2f{{y}_{1}}+c\].