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Number of common tangents
I:$$x^{2}+y^{2}=4$$
$$x^{2}+y^{2}-8x+12=0$$
a) $$0$$

II:$$x^{2}+y^{2}=1$$
$$x^{2}+y^{2}-2x-6y+6=0$$
b) $$1$$

III:$$x^{2}+y^{2}=16$$
$$x^{2}+y^{2}-8x+6y-56=0$$
c) $$4$$

IV: $$x^{2}+y^{2}-2x-6y+9=0$$
$$x^{2}+y^{2}+6x-2y+1=0$$
d)$$ 3$$



A
a,b,c,d
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B
d,c,b,c
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C
c,b,a,d
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D
d,c,b,a
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Solution

The correct option is A $$d,c,b,c$$
$$I.\ S_{1}:x^{2}+y^{2}=4$$  and  $$S_{2}:x^{2}+y^{2}-8x+12=0$$
      $$C_{1}\equiv (0,0)$$                     $$C_{2}\equiv (4,0)$$
       $$r_{1}=2$$                             $$r_{2}=2$$
$$S_{1}$$ and $$S_{2}$$ Touch each other externally, So no of common tangent are $$3.$$
$$II.\ S_{1}:x^{2}+y^{2}=1$$  and  $$S_{2}:x^{2}+y^{2}-2x-6y+6=0$$
     $$C_{1}\equiv (0,0)$$                 $$C_{2}\equiv (1,3)$$
     $$r_{1}=1$$                         $$r_{2}=2$$
$$C_{1}C_{2}=\sqrt{10}$$  and  $$r_{1}+r_{2}=3$$
$$C_{1}C_{2}>r_{1}+r_{2}$$
$$S_{1}$$ & $$S_{2}$$ does not intersect each other. So no of common tangents are $$4$$.
$$III.\ S_{1}:x^{2}+y^{2}=16$$  and  $$S_{2}:x^{2}+y^{2}-8x+6y+56=0$$
     $$C_{1}\equiv (0,0)$$                 $$C_{2}\equiv (4,3)$$
     $$r_{1}=4$$                         $$r_{2}=9$$
$$C_{1}C_{2}=5$$   and   $$r_{1}+r_{2}=13$$
$$r_{2}-r_{1}=5$$
$$C_{1}C_{2}=r_{2}-r_{1}$$
$$S_{1}$$ and $$S_{2}$$ Touches internally so no of common tangent are $$1$$.
$$IV.\ x^{2}+y^{2}-2x-6x+9=0$$ 
     $$C_{1}\equiv (1,3)$$               
     $$r_{1}=1$$
$$x^{2}+y^{2}+6x-2y+1=0$$
$$C_{2}\equiv (-3,1)$$
$$r_{2}=3$$
$$C_{1}C_{2}>r_{1}r_{2}$$
$$S_{1}$$ and $$S_{2}$$ are not intersection so, no of common tangent is $$4.$$
So, $$d,c,b,c$$

Maths

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