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In the List-I each item contains equations of two circles, List-II contains the number of common tangents for each pair of circles given in List-I. Match the items of List-I with those of the items of List-II
The correct match is
$\text { A } \quad \text { B } \quad C \quad \text { D }$
The correct match is
Options:
The correct match is
$\text { A } \quad \text { B } \quad C \quad \text { D }$
The correct match is
Solution:
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Verified Answer
The correct answer is:
IV V III II
A. $x^2+y^2+2 x+8 y-23=0$
$\begin{aligned}
& C_1(-1,-4), r_1=2 \sqrt{10} \\
& x^2+y^2-4 x-10 y+19=0 C_2(2,5), r_2=\sqrt{10} \\
& C_1 C_2=\sqrt{3^2+9^2}=\sqrt{90}=3 \sqrt{10}=r_1+r_2=3 \sqrt{10}
\end{aligned}$
$\therefore 3$ tangents
B. $x^2+y^2=1, C_1(0,0), r_1=1$
$\begin{aligned}
& x^2+y^2-2 x-6 y+6=0, C_2(1,3), r_2=2 \\
& C_1 C_2=\sqrt{1+9}=\sqrt{10}>r_1+r_2>3
\end{aligned}$
$\therefore 4$ tangents
C.
$\begin{aligned}
& \text { C. } x^2+y^2-8 x+2 y=0, C_1(4,-2), r_1=\sqrt{17} \\
& x^2+y^2-2 x-16 y+25=0, C_2(1,8), r_2=2 \sqrt{10} \\
& C_1 C_2=\sqrt{(3)^2+(10)^2}=\sqrt{9+100}=\sqrt{109} < \sqrt{17}+2 \sqrt{10}
\end{aligned}$
2 tangent
D.
$\begin{aligned}
& y^2=4, C_1(0,0), r_1=2 \\
& x^2+y^2-2 x=0, C_2(1,0), r_2=1 \\
& C_1 C_2=1,\left|r_1-r_2\right|=1 \Rightarrow C_1 C_2=\left|r_1-r_2\right|
\end{aligned}$
$\therefore$ One common tangents
$\begin{aligned}
& C_1(-1,-4), r_1=2 \sqrt{10} \\
& x^2+y^2-4 x-10 y+19=0 C_2(2,5), r_2=\sqrt{10} \\
& C_1 C_2=\sqrt{3^2+9^2}=\sqrt{90}=3 \sqrt{10}=r_1+r_2=3 \sqrt{10}
\end{aligned}$
$\therefore 3$ tangents
B. $x^2+y^2=1, C_1(0,0), r_1=1$
$\begin{aligned}
& x^2+y^2-2 x-6 y+6=0, C_2(1,3), r_2=2 \\
& C_1 C_2=\sqrt{1+9}=\sqrt{10}>r_1+r_2>3
\end{aligned}$
$\therefore 4$ tangents
C.
$\begin{aligned}
& \text { C. } x^2+y^2-8 x+2 y=0, C_1(4,-2), r_1=\sqrt{17} \\
& x^2+y^2-2 x-16 y+25=0, C_2(1,8), r_2=2 \sqrt{10} \\
& C_1 C_2=\sqrt{(3)^2+(10)^2}=\sqrt{9+100}=\sqrt{109} < \sqrt{17}+2 \sqrt{10}
\end{aligned}$
2 tangent
D.
$\begin{aligned}
& y^2=4, C_1(0,0), r_1=2 \\
& x^2+y^2-2 x=0, C_2(1,0), r_2=1 \\
& C_1 C_2=1,\left|r_1-r_2\right|=1 \Rightarrow C_1 C_2=\left|r_1-r_2\right|
\end{aligned}$
$\therefore$ One common tangents
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