Question

If P,Q and R are any three points on the curve whose equation is xy=c² then prove that the orthocentre of the triangle PQR also lies on that curve. (wirhout using parabola.)

Answer 1

Hello, It is an important result that the orthocentre of a triangle whose points lies on the hyperbola will also lie on the hyperbola Proof: Let A=(p,1/p), B=(q,1/q), C=(r,1/r) be the three points of the triangle, let H be the orthocenter. (HERE the eq is xy =c^2 but c can be assumed as1 for simplicity)

Verify that the slope of AB is exactly 1/pq , so the slope of the altitude HC is pq . Similarly, the slope of the altitude HA is qr .

Now you can calculate the position of H , by intersectingHA and HC : We have

A+x⋅(1/qr)=H=C+y⋅(1/pq)

for some x and y . You can solve this linear equation system for x and y and get that H=(−1/pqr,−pqr) . This point clearly lies on your given hyperbola.