A tangent to the ellipse x
2/9 + y
2/2 = 1 intersects the circle x
2 + y
2 = 9 at the points P and Q. It is known that R is a point on the circle x
2+ y
2 = 9. Each of P, Q and R are located
above the
x-axis.
For example, the coordinates of R cannot be (3,0), since (3,0) is not located above the x axis.
Determine the maximum area of the triangle PQR.
Draw the perpendicular bisector of tangent chord PQ. Let the intersections of the bisector with the circle be point S above x-axis and point T below the x-axis.
If point R varies freely over the entire circle then the area of PQR is zero at P or Q and increases as R approaches S or T. Local maxima of the area of PQR occur when R is coincident with S or T.
When R is restricted to have a nonnegative x-coordinate then the closest R can get to T are the endpoints of the diameter on the x-axis. Then there are three possibilities for the maximum area PQR: point S, point (-3,0), and point (3,0).
Thoughts
