91Ó°ÊÓ

Three equal circles each of radius \(r\) touch one another. The radius of the circle touching all the three given circles \(D E F\) is \(\begin{array}{ll}\text { (a) }(2-\sqrt{3}) r & \text { (b) } \frac{(2+\sqrt{3})}{\sqrt{3}} r\end{array}\) (c) \(\frac{(2-\sqrt{3})}{\sqrt{3}} r\) (d) \((2-\sqrt{3}) r\)

A man standing between two vertical posts finds that the angle subtended at his eyes by the tops of the posts is a right angle. If the heights of the two posts are two times and four times the height of the man, and the distance between them is equal to the length of the longer post, then the ratio of the distances of the man from the shorter and the longer post is (a) \(3: 1\) (b) \(2: 3\) (c) \(3: 2\) (d) \(1: 3\)

A: Orthocentre divides the altitude \(\mathrm{AD}\) in ratio \(\mathrm{AH}\) : HD \(:: \tan B+\tan C: \tan A\) R: \(\mathrm{A} H=2 R \cos A, \mathrm{H} D=2 R \cos B \cos \mathrm{C}\) and \(\frac{H A}{H D}=\left(\frac{\sin (B+C)}{\cos B \cos C}\right) / \tan A\) (a) both \(A \& \mathrm{R}\) are true \(\& \mathrm{R}\) explains \(A\) correctly (b) both \(A\) and \(\mathrm{R}\) are true but \(\mathrm{R}\) does not explains \(A\) correctly (c) \(\mathrm{A}\) is true but \(\mathrm{R}\) is false (d) \(\mathrm{A}\) is false but \(\mathrm{R}\) is true

If exradii of a \(\triangle A B C\) are 3,4 and 6, then its inradius is (a) \(\frac{3}{4}\) (b) \(\frac{4}{3}\) (c) \(\frac{5}{6}\) (d) None of these

A6-ft-tall man finds that the angle of elevation of the top of a 24 -ft-high pillar and the angle of depression of its base are complementary angles. The distance of the man from the pillar is (a) \(2 \sqrt{3} \mathrm{ft}\) (b) \(8 \sqrt{3} \mathrm{ft}\) (c) \(6 \sqrt{3} \mathrm{ft}\) (d) none of these.