Math, asked by shintesajyothapark, 1 year ago

If 4x2+y2=40,xy=6 find 9x2+4y2

adamsyakir: are you sure about that ? 4x2+y2=40 ?

Answers

Answered by adamsyakir

$4x^2+y^2=40 \\\ (2x-y)^2+4xy=40 \Rightarrow x.y=6 \\\ (2x-y)^2+4.6=40 \\\ (2x-y)^2=40-24 \\\ (2x-y)^2=16 \\\ 2x-y=\sqrt {16} \\\ 2x-y =4 \\\\ y=4+2x \Rightarrow x.y=6 \\\ x.(4+2x)=6 \\\ 4x+2x^2=6 \\\ 2x^2+4x-6=0 \\\ x^2+2x-3=0 \\\ (x+3)(x-1) =0 \\\ x=-3~v~x=1 \\\\ x=-3 \\\ y=4+2(-3) =4-6 =-2 \\\\ x=1 \\\ y=4+2(1) =4+2 =6 \\\\ \mathrm{there~are~2~points:(-3,-2)~and~(1,6)} \\\\ (-3,-2) \\\ 9x^2+4y^2=9(-3)^2+4(-2)^2 \\\ ~~~~~~~~~~~~~~=9(9)+4(4) \\\ ~~~~~~~~~~~~~~=81+16 \\\ ~~~~~~~~~~~~~~=97$

$(1,6) \\\ 9x^2+4y^2=9(1)^2+4(6)^2 \\\ ~~~~~~~~~~~~~~=9(1)+4(36) \\\ ~~~~~~~~~~~~~~=9+144 \\\ ~~~~~~~~~~~~~~=153$

kvnmurty: there are 4 solutions actually: (x, y) = (1, 6) ;; (-1,-6) ;; (3, 2) ;; (-3, -2)

adamsyakir: hmm , if using latex so lazy to correct them , and instead found a different result .

adamsyakir: Finish

kvnmurty: first do on your notes..pen paper.. then write only important steps in the answer here. so it saves time and allows easy corrections.

Answered by kvnmurty

4 x² + y² = 40 x y = 6 ⇒ y = 6/x

⇒ 4 x² + 36/x² - 40 = 0
⇒ x⁴ - 10 x² + 9 = 0

⇒ x² = 1 or 9
⇒ y² = 6²/x² = 36 or 4

⇒ 9 x² + 4 y² = 153 or 97

kvnmurty: :)

adamsyakir: Sir , how can change it .. x⁴ - 10 x² + 9 = 0 into x² = 1 or 9 ?

kvnmurty: factorization ..

Previous Question

Next Question