Multiple choice

Two circles of radii $x$ and $y$ touch each other externally and $ST$ is their common tangent, touching them at $S$ and $T$. Then $ST$ $^2$ equals -

  1. $x-y$
  2. $x+y$
  3. $2xy$
  4. $4xy$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two circles of radii x and y touching externally, the distance between centers is x + y. The common tangent length ST is given by sqrt(d^2 - (x-y)^2) = sqrt((x+y)^2 - (x-y)^2) = sqrt(4xy) = 2*sqrt(xy). Thus, ST^2 = 4xy.

AI explanation

Let the centers be C and D, with radii x and y. Drop a perpendicular from C to the radius DT, meeting it at R. In the right-angled triangle CRT, CR equals x minus y and CD equals x plus y. Using the Pythagorean theorem, ST squared equals CD squared minus CR squared, which is (x plus y) squared minus (x minus y) squared. Expanding both terms gives 4xy.

Keep practicing — related questions