Question

What are the greatest and smallest values that the function f(x, y)=xy takes on the ellipse (x^2)/8+(y^2)/2=1?

Answer 1

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Answer 2

we know that equation of ellipse x^2/a^2 + y^2/b^2 = 1

where (a,0),(- a,0),(0,b)and (-b ,0) are four point on the axes

Given

function

f(x,y)= xy

equation of ellipse

x^2/8 + y^2/2 = 1

the above equation also be written in ellipse form as

(X/sqrt8)^2 + (y/sqrt2)^2 = 1

we get point as ( sqrt8 ,0) ,(- sqrt8 - 0) ,(0 , sqrt2) and (- sqrt 2 ,0)

since line intersect the ellipse so the equation satisfied by its point

largest point

f(8,2)= 8*2 =16

smallest point

f( 0 ,-2) = 0