question number 6 6 a Show that the set S of all y such tha

question number 6

6. a. Show that the set S of all (. y) such that 2 +xyy 3 is closed b. Show that S is bounded. (Hint: Consider the equivalent equation (x+y/2)2+3v2/4 = 3. Hence. 3y-/4-3, so that-2

Solution

6 . (A) given equation : x²+xy+y²=3

=> (1/3) x² + (1/3) xy + (1/3) y² = 1

=> Ax² + B xy + C y² = 1 with A = B = C = 1/3 ( by comparing with general equation of ellipse also center (h,k) is (0,0) )

which is the equation of elipse with center at origin , so all solution ( x,y) satisfying the equation lies on ellipse centered at ( 0 , 0 )  

=> the set S of all (x,y) satisfying the equation are points of the elipse which shows the set S is closed.

( B ) as shown above the Solution of equation are points of the ellipse => set S is bounded .

(C) hint : the eqaution here is a equation of a hyperbola , and hence can be shown its unbounded