Find the acute angle between the tangents which can be drawn from (2,3) to the hyperbolas x^2 - y^2 = 2. I've found dy/dx = x/y but I don't know what to do with it. Do I simply sub in (2,3)? If so, what do I do next???
Well, you certainly don't substitute x= 2, y= 3 into the equation of the hyperbola because (2, 3) is NOT on the hyperbola! At any point, , the slope of the tangent line is given by so the equation of a line that is tangent to the hyperbola at is .
Now, since that tangent line must pass through (2, 3), you can replace x by 2 and y by 3 in that: . Multiplying through by to get rid of the fraction, or , a quadratic equation in and . Since is on the hyperbola, you also have so that and then . You can substitute that into either of the previous equations to get a quadratic equation for which, hopefully, will have two roots giving the two points where the two tangents through (2, 3) touch the hyperbola. That, of course, will give you the slopes, , of the two lines and you can get the angle between them from that.