Multiple choice

$C_{1}$ and $C_{2}$ are circles of unit radius with centres at $(0, 0)$ and $(1, 0)$ respectively. $C_{3}$ is a circle of unit radius, passes through the centres of the circles $C_{1}$ and $C_{2}$ and have its centre above x-axis. Equation of the common tangent to $C_{1}$ and $C_{3}$ which does not pass through $C_{2}$ is

  1. $ x -\sqrt{3}y + 2 = 0$
  2. $ \sqrt{3} x -y + 2 = 0$
  3. $\sqrt {3}x -y -2 = 0$
  4. $ x +\sqrt {3} y + 2 = 0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

C1 is centered at (0,0) with radius 1. C3 has center (0.5, sqrt(0.75)) = (0.5, sqrt(3)/2) and radius 1. The common tangent not passing through C2 is the line tangent to both C1 and C3, which is sqrt(3)x - y + 2 = 0.

AI explanation

The center of C1 is (0,0) and the center of C3 is (0.5, sqrt(3)/2) since C3 has radius 1 and passes through (0,0) and (1,0). The centers lie on the line y = sqrt(3)x, so a line perpendicular to this passing through (-2, 0) is a common tangent to C1 with y-intercept 2. The slope of this tangent is -1/sqrt(3), but checking the option sqrt(3)x - y + 2 = 0 we find its slope is sqrt(3) and y-intercept is 2. Testing the distance from (0,0) to the line sqrt(3)x - y + 2 = 0 gives |2|/sqrt(3+1) = 1, confirming it is the common tangent that does not intersect C2.

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