$A$ pole stands vertically inside a triangular park $ABC$. If the angle of elevation of the top of the pole from each corner of the park is the same,then in the $\Delta ABC$,the foot of the pole is at the

$A$ man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes $18 \text{ min}$ for the angle of depression of the car to change from $30^\circ$ to $45^\circ$,then after this,the time taken (in min) by the car to reach the foot of the tower is:

If the angles of depression of the top and bottom of a short building from the top of a tall building are $30^{\circ}$ and $60^{\circ}$ respectively, then the ratio of the heights of the short and tall buildings is

Let $10$ vertical poles standing at equal distances on a straight line,subtend the same angle of elevation $\alpha$ at a point $O$ on this line and all the poles are on the same side of $O$. If the height of the longest pole is $h$ and the distance of the foot of the smallest pole from $O$ is $a$,then the distance between two consecutive poles is:

Two men are on the opposite sides of a tower. They measure the angles of elevation of the top of the tower as $45^{\circ}$ and $30^{\circ}$ respectively. If the height of the tower is $40 \, m,$ find the distance between the men.

$A$ pole stands vertically inside a triangular park $\Delta ABC$. Let the angle of elevation of the top of the pole from each corner of the park be $\frac{\pi}{3}$. If the radius of the circumcircle of $\Delta ABC$ is $2$,then the height of the pole is equal to: