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Q.
Total number of lines touching atleast two circles of the family of four circles $x^{2}+y^{2}\pm3x\pm8y=0$ is
NTA AbhyasNTA Abhyas 2020Conic Sections
Solution:
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.
