Question

2x²+3xy-y²=0 , then prove that x varies as y

Answer 1

we have to prove that, x ∝ y

this is homogeneous equation. let's try to break in simpler form.

2x² + 3xy - y² = 0

dividing by y² from both sides,

⇒2x²/y² + 3xy/y² - y²/y² = 0

⇒2(x/y)² + 3(x/y) - 1 = 0

let (x/y) = m

then , 2m² + 3m - 1 = 0 now it is in the form of quadratic equation.

using formula, a = {-b ± √(b² - 4ac)}/2a

m = {-3 ± √(3² + 8)}/2(2)

= {-3 ± √17}/4

so, m = (-3 + √17)/4 , (-3 - √17)/4

or, y/x = (-3 + √17)/4, (-3, -√17)/4

or, y = {(-3 + √17)/4} x , {(-3 -√17)/4}x

hence, it is clear that, y ∝ x or, x ∝ y

also read similar questions : if x+y+1=0 prove that x²+y²+1=3xy

https://brainly.in/question/1240064

If x+y-1=0 then prove that x cube + y cube +3xy =1

https://brainly.in/question/1203861

Answer 2

Answer:

We have,

x2+y2=t−t/1  and   x4+y4=t2+t2/1

⇒(x2+y2)2=(t−t/1)2

⇒x4+y4+2x2y2=t2+t2/1−2

⇒x4+y4+2x2y2=x4+y4−2      [given]

⇒2x2y2=−2

⇒x2y2=−1

⇒y2=−x21

⇒y