MCQ
If ${a^2},\,{b^2},\,{c^2}$ be in $A.P.$, then $\frac{a}{{b + c}},\,\frac{b}{{c + a}},\,\frac{c}{{a + b}}$ will be in
- ✓$A.P.$
- B$G.P.$
- C$H.P.$
- DNone of these
Then ${b^2} - {a^2} = {c^2} - {b^2}$
$ \Rightarrow $ $(b - a)(b + a) = (c - b)(c + b)$
==> $\frac{{b - a}}{{c + b}} = \frac{{c - b}}{{b + a}}$
==> $\frac{{(b - a)(a + b + c)}}{{(c + a)(b + c)}} = \frac{{(c - b)(a + b + c)}}{{(a + b)(c + a)}}$
$ \Rightarrow $ $\frac{{{b^2} + bc - ac - {a^2}}}{{(c + a)(b + c)}} = \frac{{{c^2} + ac - ab - {b^2}}}{{(a + b)(c + a)}}$
$ \Rightarrow $ $\frac{b}{{c + a}} - \frac{a}{{b + c}} = \frac{c}{{a + b}} - \frac{b}{{c + a}}$
Hence $\frac{a}{{b + c}},\;\frac{b}{{c + a}},\;\frac{c}{{a + b}}$ be in $A.P.$
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$(\bar{z})^2+\frac{1}{z^2}$
are integers, then which of the following is/are possible value($s$) of $|z|$ ?
$\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2 y}{x+y+z},$
then which of the following statements is/are $TRUE$?
$(A)$ $2 Y = X + Z$ $(B)$ $Y=X+Z$ $(C)$ $\tan \frac{x}{2}=\frac{x}{y+z}$ $(D)$ $x^2+z^2-y^2=x z$