Question

i) The circles $x^2+y^2-8x+6y+21=0$,
$x^2+y^2+4x-10y-115=0$
touch externally.
ii) The circles $x^2+y^2-4x-6y-12=0,x^2+y^2+6x-2y+1=0$ intersect each other.
Which of the statement is correct.

• A. Only i
• B. Only ii
• C. Both i & ii
• D. Neither i nor ii

Answer 1

The correct option is B Only ii

[ From point (1) ]
$i)S_1:x^2+y^2-8x+6y+21=0$
$C_1\equiv$ and $r_1=2$
$S_2:x^2+y^2+4x-10y-115=0$
$C_1:(-2,5)$ and $r_2=12$
$\therefore C_1{C}_2=10$ and $r_2-R_1=10$
$C_1{C}_2=r_2-R_1$
$\therefore S_1$ and $S_2$ touches each other internally.
So, i is false
[ From point (1) ]
$ii)S_1:x^2+y^2-4x-6y-12=0$
$C_1\equiv (2,3)$ and $r_1=5$
$S_2:x^2+y^2+6x-2y+1=0$
$C_2\equiv (-3,1)$ and $r_2=3$
$\therefore C_1{C}_2=\sqrt{29}$ and $r_1-r_2=8$
$r_1-r_2=2$
$\Rightarrow r_1+r_2>C_1{C}_2$
So, $S_1$ and $S_2$ intersects each other ii is true.