Let A and B be two points on the hyperbola xy=1, and let C be the reflection of B through the origin.
(a) Show that C is on the hyperbola.
(b) Let Γ be the circumcircle of triangle ABC and let A' be the point on Γ diametrically opposite A. Show that A' is also on the hyperbola xy=1.
Sorry, I submitted the question before writing some things. Here's what I don't understand: what does it mean when C is the reflection of B through the origin? I thought that points can only be reflected through lines?
reflection through the origin takes (x,y) --> (-x,-y)
Clearly (-x)(-y) = xy
The hyperbola is symmetric about the origin.
google the topic and you will find more discussions and illustrations.
Thanks, I solved (a).
I also almost solved (b), just one thing I don't get: what do we do once we have found the coordinates of A'?
Oh never mind, I have finished the problem.
How do we find the coordinates of A'?
To solve these problems, we need to understand the properties of reflection and circumcircle.
(a) Show that C is on the hyperbola:
To demonstrate that C is on the hyperbola xy=1, we first need to find the coordinates of point C.
Let's assume the coordinates of point B are (x1, y1). The reflection of B through the origin (0, 0) will result in point C, which will have coordinates (-x1, -y1).
Since B lies on the hyperbola xy=1, we can substitute the coordinates of B into the equation and check if it holds true:
(x1)(y1) = 1
Now, let's substitute the coordinates of C into the equation and see if it still holds true:
(-x1)(-y1) = 1
When we simplify the equation, we get:
x1*y1 = 1
This matches the equation of the hyperbola xy=1, indicating that point C (-x1, -y1) lies on the hyperbola as well.
(b) Show that A' is also on the hyperbola:
To prove that A' is on the hyperbola xy=1, we need to use the properties of the circumcircle of triangle ABC.
The circumcircle of a triangle is the circle passing through all three vertices of the triangle. In this case, triangle ABC has vertices A, B, and C.
First, we need to find the coordinates of point A'. To do this, we can use the fact that A' is the point on the circumcircle diametrically opposite A. This means that the line passing through A and A' will pass through the center of the circumcircle.
Let's assume the coordinates of the center of the circumcircle are (h, k). The midpoint of the line segment AA' will be the average of their x-coordinates and y-coordinates. Given that the coordinates of point A are (x2, y2), we can calculate the coordinates of A' as:
Coordinates of A' = 2(h, k) - (x2, y2)
Once we have the coordinates of A', we can substitute them into the equation of the hyperbola xy=1 and check if it holds true.
(x2)(y2) = 1
(x2)(y2) = 1
This confirms that A' also lies on the hyperbola xy=1.
Therefore, we have shown that both C (reflection of B through the origin) and A' (point on the circumcircle diametrically opposite A) lie on the hyperbola xy=1.