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જો સમીકરણો $ax^2 + bx + c = 0$ અને $px^2 + qx + r = 0$, ના બીજ અનુક્રમે $\alpha_1, \alpha_2$ અને $\beta_1, \beta_2$ હોય, તો સમીકરણોની પદ્ધતિ (Syteam of Linear Equatioin ) $\alpha_1y + \alpha_2z = 0$ અને $\beta_1y + \beta_2z = 0$ શૂન્યેતર ઉકેલ ધરાવે તો શું થાય ?
$p^2br = a^2qc$
$b^2pr = q^2ac$
$r^2pb = c^2ar$
આપેલ પૈકી એકપણ નહિ.
Solution
${\alpha _1}\, + \,{\alpha _2}\, = \,\,\frac{{ – b}}{a},\,\,{\alpha _1}{\alpha _2}\, = \,\,\frac{c}{a}$
${\beta _1}\, + \,\,{\beta _2}\,\, = \,\,\frac{{ – q}}{p},\,\,{\beta _1}{\beta _2}\, = \,\,\frac{r}{p}$
જ્યાં સમીકરણો ${\alpha _{\rm{1}}}y\,\, + \,\,{\alpha _2}z\,\, = \,\,0$ અને
$\,{\beta _1}y\,\, + \,\,{\beta _2}z\,\, = \,\,0$ શૂન્યેતર ઉકેલ ધરાવે છે.
$\therefore \,\left| {\begin{array}{*{20}{c}}
{{\alpha _1}}&{{\alpha _2}}\\
{{\beta _1}}&{{\beta _2}}
\end{array}} \right|\, = \,\,0\,\,$
$ \Rightarrow \,{\alpha _1}{\beta _2}\,\, – \,\,\,{\alpha _2}{\beta _1}\, = \,0$
$\frac{{{\alpha _1}}}{{{\beta _1}}}\,\, = \,\,\frac{{{\alpha _2}}}{{{\beta _2}}}\,\, \Rightarrow \,\,\frac{{{\alpha _1}\, + \,{\alpha _2}}}{{{\beta _1} + {\beta _2}}}\, = \,\, \mp \,\sqrt {\left\{ {\frac{{{\alpha _1}\,{\alpha _2}}}{{{\beta _1}{\beta _2}}}} \right\}} $
$\therefore \,\,\,\frac{{\frac{{ – b}}{a}}}{{\frac{{ – q}}{p}}}\, = \,\sqrt {\left\{ {\frac{{\frac{c}{a}}}{{\frac{r}{p}}}} \right\}} \,$
$ \Rightarrow \,\frac{{{b^2}\,{p^2}}}{{{q^2}{a^2}}}\,\, = \,\,\frac{{cp}}{{ar}}\,$
$ \Rightarrow \,{b^2}pr\,\, = \,\,{q^2}\,ac$
