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Circle T intersects the hyperbola y=1/x at (1,1), (3, 1/3), and two other points.
What is the product of the y coordinates of the other two points? Please write in proof format. Thank you.
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The equation of the circle is
= ,
for some "a", "b" and "r".
The equation of the hyperbola is y = (given !)
You will get the equation for common points (intersection points) if you substitute equation (2) into the equation (1).
You will get
+ = , or
+ - + = 0.
Next multiply both sides by to rid of denominators. You will get
= 0, or, ordering by descending degrees of x
= 0.
The last equation is the 4-th degree equation. Its roots are x-coordinates of the common (intersection) points.
Two of the roots are given: they are x-coordinates of the given intersection points x= 1 and x= 3.
Two other roots are not known.
But, according to the Vieta's theorem for the equation of the degree 4, the product of four roots is the constant term
( ! - it is the KEY idea ! ).
Thus, = = 1, (1)
which implies
= = 3. (2)
The problem asks about , but it is simply
= . = =
due to (2).
So, the problem is solved and the answer is: the product of y-coordinates of the two other intersection points is .
Answer. The product of y-coordinates of the two other intersection points is .
Solved.
For Vieta's Theorem see this Wikipedia article.