If $p_1, p_2, p_3$ are the altitudes and $a=4, b=5, c=6$ are the sides of a triangle $ABC$,then $\frac{1}{p_1^2} + \frac{1}{p_2^2} + \frac{1}{p_3^2} =$

If in a triangle the angles are in $A.P.$ and $b:c = \sqrt{3}:\sqrt{2}$,then $\angle A$ is equal to .....$^o$.

In a $\triangle ABC$,if $\sin^2 B = \sin A$ and $2 \cos^2 A = 3 \cos^2 B$,then the triangle is

For a $\Delta ABC$,if $a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = \frac{3b}{2}$,then the sides $a, b, c$ are in:

If in a triangle $ABC$,$a\cos^2\frac{C}{2} + c\cos^2\frac{A}{2} = \frac{3b}{2}$,then its sides will be in

$A$ tower is situated on a horizontal plane. Two points lie on the line passing through the base of the tower,at distances $a$ and $b$ from the base. The angles of elevation of the top of the tower from these points are $\alpha$ and $90^\circ - \alpha$. If the line segment joining the two points subtends an angle $\theta$ at the top of the tower,find the height of the tower.