In a triangle ABC- r1-r2-r3 are the ex-radii of the ex-circles- If a-r1-r2-r3r and b-r1r2r3r3- where r is the inradius then the minimum value of amin3 tan 2A- bmin9 2A is

In any triangle ABC, nbsp; 1r=1r_1+1r_2+1r_3 r_1+r_2+r_33≥31r_1+1r_2+1r_3=3r nbsp; nbsp; nbsp; Using A.M.≥H.M r_1+r_2+r_3r≥ 9 Minimum of r_1+r_2+r_ ...

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Byju's Answer

Standard XII

Mathematics

Solution of Triangle

In a triangle...

Question

In a triangle ABC, $r_{1}, r_{2}, r_{3}$ are the ex-radii of the ex-circles. If $a = \frac{r_{1} + r_{2} + r_{3}}{r} and b = \frac{r_{1} r_{2} r_{3}}{r^{3}},$ where r is the inradius then the minimum value of $((\frac{a_{min}}{3}) {tan}^{2} A + (\frac{b_{min}}{9}) {cot}^{2} A)$ is

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Solution

In any triangle ABC, $\frac{1}{r} = \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}}$

$\frac{r_{1} + r_{2} + r_{3}}{3} \geq \frac{3}{\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}}} = 3 r$ $Using A.M. \geq H.M$

$\frac{r_{1} + r_{2} + r_{3}}{r} \geq 9$

Minimum of $\frac{r_{1} + r_{2} + r_{3}}{r} = 9$

$a_{min} = 9$

Similarly, $(r_{1} r_{2} r_{3})^{1 / 3} \geq \frac{3}{\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}}}$ $Using G.M. \geq H.M$

$\frac{r_{1} r_{2} r_{3}}{r^{3}} \geq 27 \Rightarrow b_{min} = 27$

$(\frac{a_{min}}{3}) {tan}^{2} A + (\frac{b_{min}}{9}) {cot}^{2} A = 3 {tan}^{2} A + 3 {cot}^{2} A$

$= 3 ({tan}^{2} A + {cot}^{2} A)_{min} = 6,$

$\Rightarrow$ Minimum value of the expression is 6.

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Similar questions

Q. If the sides of a triangle

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are in A.P as well as in G.P, then the value of

(\frac{r_{1}}{r_{2}} - \frac{r_{2}}{r_{3}})

is (where

r_{1}, r_{2}

and

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are the radii of ex-circles):

Q. In a triangle

A B C,

if the radii of ex-circles

r_{1}, r_{2}

and

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are given by

r_{1} = 8, r_{2} = 12

and

r_{3} = 24,

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Q. Assertion :In a

△ A B C

, if

a < b < c

and

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r_{1}, r_{2}, r_{3}

are the axradii opposite to angle

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r < r_{1} < r_{2} < r_{3}

Reason:

△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = \frac{r_{1} r_{2} r_{3}}{r}

Q. If

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is equal to

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a (r r_{1} + r_{2} r_{3}) = b (r r_{2} + r_{3} r_{1}) = c (r r_{3} + r_{1} r_{2})

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In a triangle ABC, r1,r2,r3 are the ex-radii of the ex-circles. If a=r1+r2+r3r and b=r1r2r3r3, where r is the inradius then the minimum value of ((amin3)tan2A+(bmin9)cot2A) is

In a triangle ABC, $r_{1}, r_{2}, r_{3}$ are the ex-radii of the ex-circles. If $a = \frac{r_{1} + r_{2} + r_{3}}{r} and b = \frac{r_{1} r_{2} r_{3}}{r^{3}},$ where r is the inradius then the minimum value of $((\frac{a_{min}}{3}) {tan}^{2} A + (\frac{b_{min}}{9}) {cot}^{2} A)$ is