Let $p, q$ and $r$ be the altitudes of a triangle with area $S$ and perimeter $2t$. Then, the value of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$ is

The upper $\frac{3}{4}$th portion of a vertical pole subtends an angle $\tan^{-1}\left(\frac{3}{5}\right)$ at a point in the horizontal plane through its foot and at a distance $40 \ m$ from the foot. $A$ possible height of the vertical pole is $....... \ m$.

In a $\triangle ABC$,the expression $\frac{a}{\tan A} + \frac{b}{\tan B} + \frac{c}{\tan C}$ is equal to:

If in a triangle $ABC$,$AB=5$ units,$\angle B=\cos ^{-1}\left(\frac{3}{5}\right)$ and the radius of the circumcircle of $\triangle ABC$ is $5$ units,then the area (in sq. units) of $\triangle ABC$ is:

In a triangle $ABC$,$\angle B = \frac{\pi}{3}$ and $\angle C = \frac{\pi}{4}$,and $D$ divides $BC$ internally in the ratio $1 : 3$. Then $\frac{\sin \angle BAD}{\sin \angle CAD}$ is equal to

If $a \cos A = b \cos B$,then $\Delta ABC$ is