Giải hệ phương trình : $\left\{\begin{array}{l}\frac{x+\sqrt{x^2-y^2}}{x-\sqrt{x^2-y^2}}=\frac{10x}{12}\\\frac{x}{y}=\frac{4x+26}{24-13y}\end{array}\right.$
Giải hệ phương trình : $\left\{\begin{array}{l}\frac{x+\sqrt{x^2-y^2}}{x-\sqrt{x^2-y^2}}=\frac{10x}{12}\\\frac{x}{y}=\frac{4x+26}{24-13y}\end{array}\right.$
Giải thích các bước giải:
Ta có :
$\dfrac{x}{y}=\dfrac{4x+26}{24-13y}$
$\to y=\dfrac{24x}{17x+26}$
$\to \dfrac{x+\sqrt{x^2-(\dfrac{24x}{17x+26})^2}}{x-\sqrt{x^2-(\dfrac{24x}{17x+26})^2}}=\dfrac{10x}{12}$
$\to \left(x+\sqrt{x^2-\left(\frac{24x}{17x+26}\right)^2}\right)\cdot \:12=\left(x-\sqrt{x^2-\left(\frac{24x}{17x+26}\right)^2}\right)\cdot \:10x$
$\to 6\sqrt{\frac{289x^4+884x^3+100x^2}{289x^2+884x+676}}+5x\sqrt{\frac{289x^4+884x^3+100x^2}{289x^2+884x+676}}-5x^2+6x=0$
$\to \sqrt{\frac{289x^4+884x^3+100x^2}{289x^2+884x+676}}\left(6+5x\right)=5x^2-6x$
$\to \sqrt{\frac{289x^4+884x^3+100x^2}{289x^2+884x+676}}=\frac{5x^2-6x}{6+5x}$
$\to \left(\sqrt{\frac{289x^4+884x^3+100x^2}{289x^2+884x+676}}\right)^2=\left(\frac{5x^2-6x}{6+5x}\right)^2$
$\to \frac{289x^4+884x^3+100x^2}{289x^2+884x+676}=\frac{25x^4-60x^3+36x^2}{36+60x+25x^2}$
$\to x^2\left(5x+6\right)^2\left(17x+2\right)\left(17x+50\right)-x^2\left(5x-6\right)^2\left(17x+26\right)^2=0$
$\to x^2\left(34680x^3+91680x^2+46560x-20736\right)=0$
$\to 24x^2\left(1445x^3+3820x^2+1940x-864\right)=0$
$\to x=0,\:x\approx \:0.27763\dots $