How do we find the range of the binary relation 4x^2 + 9y^2 = 36 ?

2 Answers
Aug 13, 2017

Set x equal to zero and then solve the equation for both values of y; the negative value will be the minimum of the range and the positive value will be the maximum.

Explanation:

Given: 4x^2 + 9y^2 = 36

Let x = 0:

4(0)^2 + 9y^2 = 36

9y^2=36

y^2=4

y = -2 and y = 2

The range is -2 <= y <= 2

Aug 13, 2017

The range is y in [-2,2]

Explanation:

The equation is

4x^2+9y^2=36

4x^2=36-9y^2

x^2=9/4(4-y^2)

x=3/2sqrt(4-y^2)

x=3/2sqrt((2-y)(2+y))

x=f(y)

x will depend on (2-y)(2+y)>=0

Let p(y)=(2-y)(2+y)

We need a sign chart

color(white)(aaaa)ycolor(white)(aaaa)-oocolor(white)(aaaaa)-2color(white)(aaaaaaaa)2color(white)(aaaaa)+oo

color(white)(aaaa)2+ycolor(white)(aaaaaa)-color(white)(aa)0color(white)(aaaa)+color(white)(aaaaa)+

color(white)(aaaa)2-ycolor(white)(aaaaaa)+color(white)(aaaaaaa)+color(white)(aa)0color(white)(aa)-

color(white)(aaaa)p(y)color(white)(aaaaaaa)-color(white)(aa)0color(white)(aaaa)+color(white)(aa)0color(white)(aa)-

Therefore,

p(y)>=0 when y in [-2,2]

graph{4x^2+9y^2=36 [-9.22, 8.56, -3.235, 5.654]}