Абсциссы точек касания
![x_1,x_2](https://tex.z-dn.net/?f=x_1%2Cx_2)
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Угловые коэфф. касательных
![k_1=y'(x_1),\; k_2=y'(x_2)](https://tex.z-dn.net/?f=k_1%3Dy%27%28x_1%29%2C%5C%3B+k_2%3Dy%27%28x_2%29)
Уравнение касательной:
![y=y(x_1)+y'(x_1)(x-x_1)](https://tex.z-dn.net/?f=y%3Dy%28x_1%29%2By%27%28x_1%29%28x-x_1%29)
![y=x^2,\; \; y(x_1)=x_1^2\\\\y'=2x,y'(x_1)=2x_1\\\\Yravn.kasat.\; \; y=x_1^2+2x_1(x-x_1)](https://tex.z-dn.net/?f=y%3Dx%5E2%2C%5C%3B+%5C%3B+y%28x_1%29%3Dx_1%5E2%5C%5C%5C%5Cy%27%3D2x%2Cy%27%28x_1%29%3D2x_1%5C%5C%5C%5CYravn.kasat.%5C%3B+%5C%3B+y%3Dx_1%5E2%2B2x_1%28x-x_1%29)
Теперь подставим координаты точки, через которую проходит касательная, (0,-2) , в уравнение касательной вместо переменных:
![-2=x_1^2+2x_1(0-x_1)\\\\-2=x_1^2-2x_1^2,\; \; x_1^2=2,\; x_1=\sqrt2,\\\\x_2=-\sqrt2](https://tex.z-dn.net/?f=-2%3Dx_1%5E2%2B2x_1%280-x_1%29%5C%5C%5C%5C-2%3Dx_1%5E2-2x_1%5E2%2C%5C%3B+%5C%3B+x_1%5E2%3D2%2C%5C%3B+x_1%3D%5Csqrt2%2C%5C%5C%5C%5Cx_2%3D-%5Csqrt2)
В принципе мы имеем обе точки касания:
![A(\sqrt2,2),\; B(-\sqrt2,2)](https://tex.z-dn.net/?f=A%28%5Csqrt2%2C2%29%2C%5C%3B+B%28-%5Csqrt2%2C2%29)
Подставим значения абсцисс в уравнение касательной.
![a)\; \; y=2+2\sqrt2(x-\sqrt2)\; \to \; y=2+2\sqrt2x-4,\\\\y=2\sqrt2x-2\; \to k_1=2\sqrt2\\\\b)\; \; y=2-2\sqrt2(x+\sqrt2),\to \; y=-2\sqrt2x-2\; \to k_2=-2\sqrt2](https://tex.z-dn.net/?f=a%29%5C%3B+%5C%3B+y%3D2%2B2%5Csqrt2%28x-%5Csqrt2%29%5C%3B+%5Cto+%5C%3B+y%3D2%2B2%5Csqrt2x-4%2C%5C%5C%5C%5Cy%3D2%5Csqrt2x-2%5C%3B+%5Cto+k_1%3D2%5Csqrt2%5C%5C%5C%5Cb%29%5C%3B+%5C%3B+y%3D2-2%5Csqrt2%28x%2B%5Csqrt2%29%2C%5Cto+%5C%3B+y%3D-2%5Csqrt2x-2%5C%3B+%5Cto+k_2%3D-2%5Csqrt2)
Угол между прямыми можно найти по формуле
![tg \alpha =|\frac{k_1-k_2}{1+k_1k_2}|\\\\tg \alpha =|\frac{2\sqrt2-(-2\sqrt2)}{1+2\sqrt2(-2\sqrt2)}|=|\frac{4\sqrt2}{1-8}|=\frac{4\sqrt2}{7}\\\\ \alpha =arctg\frac{4\sqrt2}{7}](https://tex.z-dn.net/?f=tg+%5Calpha+%3D%7C%5Cfrac%7Bk_1-k_2%7D%7B1%2Bk_1k_2%7D%7C%5C%5C%5C%5Ctg+%5Calpha+%3D%7C%5Cfrac%7B2%5Csqrt2-%28-2%5Csqrt2%29%7D%7B1%2B2%5Csqrt2%28-2%5Csqrt2%29%7D%7C%3D%7C%5Cfrac%7B4%5Csqrt2%7D%7B1-8%7D%7C%3D%5Cfrac%7B4%5Csqrt2%7D%7B7%7D%5C%5C%5C%5C+%5Calpha+%3Darctg%5Cfrac%7B4%5Csqrt2%7D%7B7%7D)