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10-1.Circle and System of Circles
normal
If the line $x\, cos \theta + y\, sin \theta = 2$ is the equation of a transverse common tangent to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6 \sqrt{3} \,x - 6y + 20 = 0$, then the value of $\theta$ is :
A
$5\pi /6$
B
$2\pi /3$
C
$\pi /3$
D
$\pi /6$
Solution
$C_1C_2 = r_1 + r_2$
$C_1 = (0, 0) ; C_2 = (3\sqrt{3}, 3)\, \& \,r_1 = 2, r_2 = 4$
$\Rightarrow$ circle touch each other
$T.C.T = \sqrt{3} \,x + y – 4 = 0$
comparing with $x\, cos \theta + y\, sin \theta = 2$
$\theta =$ $\frac{\pi}{6}$
Standard 11
Mathematics
