The angle of elevation of an object from a point P on the level ground is $\mathrm{\alpha }$. Moving d metres on the ground towards the object, the angle of elevation is found to be $\mathrm{\beta }$. Then the height (in metres) of the object is

• $\mathrm{dtan}\left(\mathrm{\alpha }\right)$

• $\mathrm{dcot}\left(\mathrm{\beta }\right)$

• $\frac{\mathrm{d}}{\cot \left(\mathrm{\alpha }\right) + \cot \left(\mathrm{\beta }\right)}$

• $\frac{\mathrm{d}}{\cot \left(\mathrm{\alpha }\right) - \cot \left(\mathrm{\beta }\right)}$

A vertical pole subtends an angle ${\tan }^{-1}\left(\frac{1}{2}\right)$ at a point P on the ground. If the angles substended by the upper half and the lower half of the pole at P are respectively $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ , then $\left(\tan \left(\mathrm{\alpha }\right), \tan \left(\mathrm{\beta }\right)\right)$ is equal to

• $\left(\frac{1}{4}, \frac{1}{5}\right)$

• $\left(\frac{1}{5}, \frac{2}{9}\right)$

• $\left(\frac{2}{9}, \frac{1}{4}\right)$

• $\left(\frac{1}{4}, \frac{2}{9}\right)$