Question

If the length of a common internal tangent to two circles is $7$, and that of a common external tangent $11$, then the product of the radii of the two circles is:

• A. $18$
• B. $20$
• C. $16$
• D. $12$

Answer 1

The correct option is A $18$

A] given

the length at common internal

tangent to two circular is 7 & nocturnal tangent is 11

Let d be the distance between the center of circular.

Let R, r he the radii where

$R⩾r$

So,

$d^2=(R+r{)}^2+7^2$

and

$d^2=(R-r{)}^2+{11}^2$

from the always equations

$(R+r{)}^2+7^2=(R-r{)}^2+{11}^2$

$=(R+r{)}^2-(R-r{)}^2={11}^2-7^2$

$=(R^2+r^2+2Rr)-(R^2+r^2-2Rr)=121-49$

$=4Rr=72$

$Rr=18$

$\therefore$The product of radii of the

two circular is 18