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AP EAMCET Engineering 21 Apr 2019 Shift 1 Paper

Section: Mathematics

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Question : 28 of 160

Marks: +1, -0

In a ΔABC , if

\frac{2r_2r_3}{r_2-r_1}

r_3-r_1

\frac{r_1(r_2+r_3)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}

=

$\frac{a^2+b^2+c^2}{\Delta^2}$
b - c
$\frac{1}{2R}$
2R

Solution:

Consider the expression,

\; \frac{2 r_{2} r_{3}}{r_{2}-r_{1}} \;\; = r_{3}-r_{1}

2 r_{2} r_{3} \;\; = (r_{2}-r_{1})(r_{3}-r_{1})

2 \; \frac{\Delta}{s-b} \; \frac{\Delta}{s-c} \;\; = \left( \frac{\Delta}{s-b} - \frac{\Delta}{s-c} \right) \left( \frac{\Delta}{s-c} - \frac{\Delta}{s-a} \right)

\; \frac{2 \Delta^{2}}{(s-b)(s-c)} \;\; = \Delta^{2} \left\{ \; \frac{s-a-s+b}{(a-b)(s-a)} \right\} \left\{ \; \frac{s-a-s+c}{(s-c)(s-a)} \right\}

Further simplify the above,

\; \frac{2}{(s-b)(s-c)} \;\; = \; \frac{b-a}{(s-b)(s-a)} \times \; \frac{c-a}{(s-c)(s-a)}

\;\; = s(s-a)^{2} = (b-a)(c-a)

\; \frac{2(b+c-a)^{2}}{4} \;\; = (b-a)(c-a)

b^{2}+c^{2}+a^{2}+2 b c-2 c a-2 a b=2

Then

b^{2}+c^{2}+a^{2} \;\; =2 a^{2}

a^{2} \;\; =b^{2}+c^{2}

Now

\frac{r_{1}(r_{2}+r_{3})}{\sqrt{r_{1} r_{2}+r_{2} r_{3}+r_{3} r_{1}}} \;\; = \frac{ \; \frac{\Delta}{s-a} \times \Delta \left\{ \frac{1}{s-b} + \frac{1}{s-c} \right\} }{s}

\;\; = \; \frac{\Delta^{2}(2 s-b-c)}{s(s-a)(s-b)(s-c)}

\;\; = \; \frac{\Delta^{2}(a+b+c-b-c)}{\Delta^{2}}

\;\; = a

\;\; = 2 R

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