# Problem Study (Geometry applying Trigonometry and Calculus)

(a)    Let be a triangle with right angle and hypotenuse .
(See the figure.)
If the inscribed circle touches the hypotenuse at D,
show that .
(b)   If  , express the radius $r$ of the inscribed circle in terms  of $a$ and $\theta$
(c)   If  $a$ is fixed and  $\theta$ varies, find the maximum value of $r$.
Solution
Let $O$  be the centre of the circle and $E$ and $F$ be points of
tangency of $AC$ and $AB$ respectively.
(given)
Draw and

Since and , is a square.
Therefore
Let .

(a)

(b)   Draw . Since $O$ is incentre, bisects .
In right ,

In right , By Pythagoras Theorem,

(c)   Since $a$ is fixed and  $\theta$ varies, is a function of and the rate of
change of with respect to is

has stationary value when  .

and .

When ,
.
will be maximum value when .