Question

Total number of lines touching atleast two circles of the family of four circles $x^2+y^2\pm 3x\pm 8y=0$ is

• A. $8$
• B. $10$
• C. $12$
• D. $14$

Answer 1

Answer: (D)

Let, $C_1$ be circle $x^2+y^2+8x+8y=0$ ,
$C_2$ be circle $x^2+y^2+8x-8y=0$ ,
$C_3$ be circle $x^2+y^2-8x+8y=0$
& $C_4$ be circle $x^2+y^2-8x+8y=0$
Now, common tangents between $C_1\&C_2$ is $2$ ,
common tangents between $C_1\&C_3$ is $2$ ,
common tangents between $C_1\&C_4$ is $3$ ,
common tangents between $C_2\&C_3$ is $3$ ,
common tangent between $C_2\&C_4$ is $2$
& common tangent between $C_3\&C_4$ is $2$
Total $=14$

$C_1\&C_4$ touch each other, $C_2\&C_4$ touch each other, rest pair for circles intersects each other.