Multiple choice

If the direct common tangents of the circles $x^{2}+y^{2}=1$ and $(x-5)^{2}+y^{2}=4$ intersect each others at point $D$ and the transverse common tangents to them intersect each other at $T$, then $DT$ is equal to

  1. $5$ units
  2. $\dfrac{10}{3}$ units
  3. $\dfrac{20}{3}$ units
  4. $7$ units
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C Correct answer
Explanation

The centers of the circles are (0,0) and (5,0) with radii 1 and 2. The point D (intersection of direct tangents) divides the line joining centers externally in ratio r1:r2 = 1:2. D = ( (1*5 - 2*0)/(1-2), 0 ) = (-5, 0). The point T (intersection of transverse tangents) divides the line internally in ratio 1:2. T = ( (1*5 + 2*0)/(1+2), 0 ) = (5/3, 0). The distance DT = |5/3 - (-5)| = |5/3 + 15/3| = 20/3.

AI explanation

For the circles centered at (0,0) with radius 1 and (5,0) with radius 2, the intersection point T of the transverse common tangents divides the line segment between centers internally in the 1:2 ratio, making it (5/3, 0). The intersection point D of the direct common tangents divides the segment externally in the 1:2 ratio, placing it at (-5, 0). The distance DT is calculated as 5/3 - (-5) = 20/3 units.

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