Let the circle $x^2+y^2=9$ intersect the hyperbola $xy=4$ at four points. If the product of the slopes of the lines joining these points to the origin is $k$, find the value of $k$.

Solution

1. Let the intersection points be $(x_i,y_i)$ for $i=1,2,3,4$.
2. The slopes are $m_i=y_i/x_i$.
3. From $xy=4$, we have $y_i=4/x_i$, so $m_i=4/x_i^2$.
4. Substituting $y=4/x$ into $x^2+y^2=9$ gives $x^4-9x^2+16=0$.
5. The product of slopes is $$\begin{array}{rl}{m}_1{m}_2{m}_3{m}_4& =(4^4)/(x_1{x}_2{x}_3{x}_4{)}^2\\ & =256/16^2\\ & =1\end{array}$$.
6. Hence the answer is (A).