1
IIT-JEE 2000
Subjective
+7
-0
Let $$ABC$$ be a triangle with incentre $$I$$ and inradius $$r$$. Let $$D,E,F$$ be the feet of the perpendiculars from $$I$$ to the sides $$BC$$, $$CA$$ and $$AB$$ respectively. If $${r_1},{r_2}$$ and $${r_3}$$ are the radii of circles inscribed in the quadrilaterals $$AFIE$$, $$BDIF$$ and $$CEID$$ respectively, prove that $${{{r_1}} \over {r - {r_1}}} + {{{r_2}} \over {r - {r_2}}} + {{{r_3}} \over {r - {r_3}}} = {{{r_1}{r_2}{r_3}} \over {\left( {e - {r_1}} \right)\left( {r - {r_2}} \right)\left( {r - {r_3}} \right)}}$$\$
2
IIT-JEE 1999
Subjective
+10
-0
Let $$ABC$$ be a triangle having $$O$$ and $$I$$ as its circumcenter and in centre respectively. If $$R$$ and $$r$$ are the circumradius and the inradius, respectively, then prove that $${\left( {IO} \right)^2} = {R^2} - 2{\mathop{\rm Rr}\nolimits}$$. Further show that the triangle BIO is a right-angled triangle if and only if $$b$$ is arithmetic mean of $$a$$ and $$c$$.
3
IIT-JEE 1998
Subjective
+8
-0
A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose $${60^ \circ }$$ and $${30^ \circ }$$ are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points $$P$$ and $$Q$$ respectively on its path. Let $$\theta$$ be the angle of elevation of the bird when it is a point on the are of the circle exactly midway between $$P$$ and $$Q$$. Find the numerical value of $${\tan ^2}\theta$$. (Assume that the observer is not inside the vertical projection of the path of the bird.)
4
IIT-JEE 1998
Subjective
+8
-0
Prove that a triangle $$ABC$$ is equilateral if and only if $$\tan A + \tan B + \tan C = 3\sqrt 3$$.
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