analytic geometry
calculus
challenge problem
trigonometry
ဆယ္တန္းသခၤ်ာ
ဆယ္တန္းသခၤ်ာေမးခြန္း
Problem Study (Geometry applying Trigonometry and Calculus)
June 04, 2016
(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
of the inscribed circle in terms of
and
.
(c) If
is fixed and
varies, find the maximum value of
.
Solution
Let
be the centre of the circle and
and
be points of
tangency of
and
respectively.
(given)
Draw
and
.
Since
and
,
is a square.
Therefore
Let
.
(a)
(b) Draw
. Since
is incentre,
bisects
.
In right
,
In right
, By Pythagoras Theorem,
(c) Since
is fixed and
varies,
is a function of
and the rate of
change of
w
ith respect to
is
has stationary value when
.
and
.
When
,
.
will be maximum value when
.