Question

find the equation of the tangent line to the hyperbola y=3/x at the point (3,1)

Answer 1

we will find the equation in the form y=mx+b (m is slope and b is y-axis intersect)
\[m=\left[ \frac{ df(x) }{ dx } \right]_{x=3}=\left[ -\frac{ 3 }{ x^2} \right]_{x=3}=-\frac{ 1 }{ 3 }\]Now we know the line is \[y=-\frac{ 1 }{ 3}x+b\]And we know that when x=3, y=1, then\[y=-\frac{ 1 }{ 3 }·3+b=1 \rightarrow -1+b=1 \rightarrow b=2\]Then the line is:\[y=-\frac{ 1 }{ 3 }x+2\]