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Intersection And Maximum Triangle (Posted on 2008-03-30) Difficulty: 3 of 5
A tangent to the ellipse x2/9 + y2/2 = 1 intersects the circle x2 + y2 = 9 at the points P and Q. It is known that R is a point on the circle x2+ y2 = 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.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Some Thoughts Thoughts Comment 5 of 5 |
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).

  Posted by Brian Smith on 2019-11-24 11:16:42
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