Multiple choice

Circle $C_1$ has radius $2$ and circle $C_2$ has radius $3$, and the distance between the centers of $C_1$ and $C_2$ is $7$. If two lines, one tangent to both circles and other passing through the center of both circles, intersect at point $P$ which lies between centers of $C_1$ and $C_2$, the the distance between $P$ and the center of $C_1$

  1. $\dfrac{9}{4}\ units$
  2. $\dfrac{7}{3}\ units$
  3. `$\dfrac{8}{3}\ units$
  4. $\dfrac{14}{5}\ units$
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D Correct answer
Explanation

The point P divides the distance between centers in the ratio of the radii (2:3) because the tangent line creates similar triangles with the centers. The distance between centers is 7, so the distance from the center of C1 is (2/(2+3)) * 7 = 14/5.

AI explanation

Using the properties of similar triangles formed by the radii to the tangent and the line of centers, the point of intersection divides the distance in the ratio of the radii. Setting the distance x from the center of the circle with radius 2, we write x / (7 - x) = 2 / 3. Solving this equation yields 3x = 14 - 2x, so 5x = 14 and x = 14/5. The distance is 14/5 units.

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