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`        The equations of the common tangebts to the two hypebolas x^2/a^2-y^2/b^2=1 and y^2/a^2-x^2/b^2=1`
8 months ago

```							Note that hyperbola 1 is the symmetric of hyperbola 2 with line y= x as axis of symmetry.in addition , hyperbola 1 is also the symmetric of hyperbola 2 with line y= -x as axis of symmetry.so the tangent lines to both hyperbolas should be in form y= x+t , y= x-t, y=-x+t, y=-x-tassume that line y= x+ t tangent to hyperbola 1, the we havex^2/a^2-(x+t)^2/b^2= 1or (a^2-b^2)x^2+2.a^2.t.x+a^2.t^2+a^2.b^2= 0the line tangent to hyperbola 1 require that the determinant of above quadratic equation equal zero.work out in details give us t= + sqrt(a^2-b^2) or t= -sqrt(a^2-b^2)and equations of 4 tangent lines to both hyperbolas arey= x+sqrt(a^2-b^2)y=x-sqrt(a^2-b^2)y=-x+sqrt(a^2-b^2)y=-x-sqrt(a^2-b^2)
```
8 months ago
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