KVPY Sample Paper KVPY Stream-SX Model Paper-7

If a circle C passing through the point (4, 0) touches the circle \[{{x}^{2}}+{{y}^{2}}+4x-6y=12\] externally at the point \[(1,-1),\] then the radius of C is:

A) \[2\sqrt{5}\]

B) 4

C) 5

D) \[\sqrt{57}\]

Correct Answer: C

Solution :

Tangent at\[(1,-1)\]is

\[x(1)+y(-1)+2(x+1)-3(y-1)-12=0\]

\[=3x-4y=7\]

Required circle is

\[{{(x-1)}^{2}}+{{(y+1)}^{2}}+\lambda (3x-4y-7)=0\]

It passes through\[(4,0)\]

\[\Rightarrow \] \[9+1+\lambda (12-7)=0\]

\[\Rightarrow \] \[\lambda =-2\]

\[\Rightarrow \]Required circle is\[{{x}^{2}}+{{y}^{2}}-8x+10y+16=0\]

\[\Rightarrow \]Radius\[=\sqrt{16+25-16}=5\]