Quandaries and Queries


The equation of a hyperbola is 32x*2 -18Y*2-64x +72y +248=0. The equation of a tangent line to this hyperbola is y= (16/15)X + 10/3 I have been trying to find the point where this line intersects the graph. What I did was solve for x and then plugged in the result into the equation of the hyperbola, but I am getting two answers and I am supposed to get only one because the line is tangent to the graph. For this reason, I would like to know what I am doing wrong or what I have to do to know which answer is correct.



Your technique is fine, you must have an algebra error. I followed your instructions and solved the equation of the tangent line for x getting

x =16/15(y - 10/3)

I substituted this value into the equation of the hyperbola and simplified to get

9/8(26 - 3y)2 = 0

and hence y = 26/3

I find that sometimes I make the same algebra error again and again so in this situation I would try a different attack. Rather than solving the tangent line equation for x, just substitute

y = 16/15x + 10/3

into the equation of the hyperbola and solve for x. Doing it this way I got

 288/25 (x - 5)2 = 0

and hence x = 5.



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