## Angle of Intersection Between Two Curves

If two curves meet at a point, the angle between the curves is the angle between the tangent vectors to the curves.
The curves
$y=x^2-5x-1$
and
$y=2x^2+x+7$
intersect at the solution to
$x^2+x+7=x^2-5x-1 \rightarrow x^2+6x+8=0 \rightarrow (x+4)(x+2)=0$
.
The curves meet at
$x+4=0 \rightarrow x=-4, \; x+2=0 \rightarrow x=-2$
.
The gradients functions of the curves at this point are
$\frac{dy}{dx}=2x-5, \; \frac{dy}{dx}=4x-1$
and the gradients at
$x=-2$
are
$2 \times -2-5=-9$
and
$4 \times -2+1=-7$
.
The angle between the curves at
$x=-2$
is
$tan^{-1}(-7)-tan^{-1}(-7)=1,79^o$
.

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