The ex-radii `r_(1), r_(2), r_(3)` of a `DeltaABC` are in Hp, show that its sides a,b,c, are in AP.

The exradii r_1,r_2 and r_3 of A B C are in H.P. Show that its sides a , ba n dc are in AdotPdot

In a triangle, if r_(1) gt r_(2) gt r_(3), then show a gtb gt c.

If a,b,c are in AP and a^(2),b^(2),c^(2) are in HP, then

Statement I In a Delta ABC, if a lt b lt c and ri si inradius and r_(1), r_(2) ,r_(3) are the exradii opposite to angle A,B,C respectively, then r lt r_(1) lt r_(2) lt r_(3). Statement II For, DeltaABC r_(1)r_(2)+r_(2)r_(3)+r_(3)r_(1)=(r_(1)r_(2)r_(3))/(r)

Prove that if two bubbles of radii r_(1) and r_(2)(r_(1)ltr_(2)) come in contact with each other then the radius of curvature of the common surface r=(r_(1)r_(2))/(r_(2)-r_(1))

Two small metallic balls of radii R_(1) & R_(2) are kept in vacuum at a large distance compared to the radii. Find the ratio between the charges on the two balls at which electrostatic energy of the system is minimum.

If r_(1),r_(2) and r_(3) are the position vectors of three collinear points and scalars l and m exists such that r_(3)=lr_(1)+mr_(2) , then show that l+m=1.

omega is the cube root of 1 and omega ne 1 . Now r_(1), r_(2) and r_(3) are the number obtained while tossing dice thrice. Then ………… is the probability for omega^(r^(1)) +omega^(r^(2))+omega^(r^(3))=0

Prove that if two bubbles of radii r_(1) and r_(2) coalesce isothermally in vacuum then the radius of new bubble will be r=sqrt(r_(1)^(2)+r_(2)^(2))

Consider two conducting spheres of radii R_(1) and R_(2) with R_(1) gt R_(2) . If the two are at the same potential, the larger sphere bas more charge than the smaller sphere. State whether the charge density of the smaller sphere is more or less than that of the larger one.