euclidean-geometrygeometry
Two circles of radius $8$ are placed inside a semi-circle of radius $25$.The two circles are each tangent to the diameter and to the semi-circle. If the distance between the centers of the two circles is $\lambda$, then the value of $\frac{60}{\lambda}$.
My try: I am not sure if I am correct, I guess these two circles touch each other and $\lambda$ is $16$.Can someone explain me correct answer and correct method.
If by "circle" you mean the set of points in a plane equidistant from a given point (the center), then there are two possibilities. If one circle is inside the other, then you'll subtract the smaller radius from the larger, and otherwise you'll add the radii.
If by "circle" you mean the set of points in a plane equidistant from a given point (the center) and all points inside it, then we can't put one circle inside the other and have them touch only once, so we must add the two radii. (This latter type of "circle" is usually called a "disk" instead.)
Solving for the radii, we get $\{3,5,6\}$.
For the area we get $$ \begin{align} &\sqrt{14(14-8)(14-9)(14-11)}\\[9pt] &-\frac{(8+9-11)^2}8\arccos\left(\frac{8^2+9^2-11^2}{2\cdot8\cdot9}\right)\\ &-\frac{(11+8-9)^2}8\arccos\left(\frac{11^2+8^2-9^2}{2\cdot11\cdot8}\right)\\ &-\frac{(9+11-8)^2}8\arccos\left(\frac{9^2+11^2-8^2}{2\cdot9\cdot11}\right) \end{align} $$ which is $3.05537320587455$.
Best Answer
Try to figure that out from the attached diagram.