Question

Circles	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. $a,b,c,d$
• B. $d,c,b,c$
• C. $c,b,a,d$
• D. $d,c,b,a$

Answer 1

Answer: (B)

$d,c,b,c$

$I.S_1:x^2+y^2=4$ 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, $d,c,b,c$