JEE Main & Advanced Mathematics Circle and System of Circles Tangent to a Circle at a Given Point

(1) Point form    

(i) The equation of tangent at \[({{x}_{1}},{{y}_{1}})\] to circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[x{{x}_{1}}+y{{y}_{1}}={{a}^{2}}\].

(ii) The equation of tangent at \[({{x}_{1}},{{y}_{1}})\] to circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]is\[x{{x}_{1}}+y{{y}_{1}}+g(x+{{x}_{1}})+f(y+{{y}_{1}})+c=0\].

(2) Parametric form : Since parametric co-ordinates of a point on the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[(a\,\cos \theta ,\,a\,\sin \theta ),\] then equation of tangent at \[(a\,\cos \theta ,\,a\,\sin \theta )\] is \[x.\,a\,\cos \theta +y\,.\,a\,\sin \theta ={{a}^{2}}\]

or \[x\,\cos \theta +y\,\sin \theta =a\].

(3) Slope form : The straight line \[y=mx+c\] touches the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] if \[{{c}^{2}}={{a}^{2}}(1+{{m}^{2}})\] and the point of contact of tangent \[y=mx\pm a\sqrt{1+{{m}^{2}}}\] is \[\left( \frac{\mp ma}{\sqrt{1+{{m}^{2}}}},\,\,\frac{\pm a}{\sqrt{1+{{m}^{2}}}} \right)\].