Circle 2 Question 7
8. The number of common tangents to the circles $x^{2}+y^{2}-4 x-6 y-12=0$ and $x^{2}+y^{2}+6 x+18 y+26=0$ is
(2015)
(a) 1
(b) 2
(c) 3
(d) 4
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Answer:
Correct Answer: 8. (c)
Solution:
- PLAN Number of common tangents depend on the position of the circle with respect to each other.
(i) If circles touch externally $\Rightarrow C _1 C _2=r _1+r _2, 3$ common tangents.
(ii) If circles touch internally $\Rightarrow C _1 C _2=r _2-r _1, 1$ common tangent.
(iii) If circles do not touch each other, 4 common tangents.
Given equations of circles are
$$ \begin{array}{r} x^{2}+y^{2}-4 x-6 y-12=0 \\ x^{2}+y^{2}+6 x+18 y+26=0 \end{array} $$
Centre of circle (i) is $C _1(2,3)$ and radius
$$ =\sqrt{4+9+12}=5\left(r _1\right) $$
Centre of circle (ii) is $C _2(-3,-9)$ and radius
$$ =\sqrt{9+81-26}=8\left(r _2\right) $$
$$ \begin{aligned} & \text { Now, } \quad C _1 C _2=\sqrt{(2+3)^{2}+(3+9)^{2}} \\ & \Rightarrow \quad C _1 C _2=\sqrt{5^{2}+12^{2}} \\ & \Rightarrow \quad C _1 C _2=\sqrt{25+144}=13 \\ & \therefore \quad r _1+r _2=5+8=13 \\ & \text { Also, } \quad C _1 C _2=r _1+r _2 \end{aligned} $$
Thus, both circles touch each other externally. Hence, there are three common tangents.
