Question

pozzzzz Solve fast. ​

Answer 1

⛦ Answer ⛦

$\sf{e=\dfrac{\sqrt{13}}{3}}$

⛦ Step-by-step explanation ⛦

$\sf{\implies4x^2-9y^2=2ax+b^2}\\\sf{\implies4x^2-2ax-9y^2=b^2}\\\sf{\implies4\left(x^2+\dfrac{ax}{2}\right)-9y^2=b^2}\\\sf{\implies4\left(x^2-\dfrac{2ax}{4}+\dfrac{a^2}{16}-9y^2\right)=b^2-\dfrac{a^2}{4}}\\\sf{\implies4\left(x-\dfrac{a}{4}\right)^2-9y^2=b^2-\dfrac{a^2}{4}=k(say)}\\\sf{\implies\dfrac{\left(x-\dfrac{a}{4}\right)^2}{\dfrac{k}{4}}-\dfrac{y^2}{\dfrac{k}{9}}}\\\sf{\implies\:Let\:k>0}\\\sf{\implies\:Therefore,\:e=\sqrt{1+\dfrac{\dfrac{k}{9}}{\dfrac{k}{4}}}}\\\sf{\implies\:e=\sqrt{1+\dfrac{4}{9}}}\\\\\boxed{\sf{e=\dfrac{\sqrt{13}}{3}}}$

Answer 2

Answer:

⛦ Answer ⛦

\sf{e=\dfrac{\sqrt{13}}{3}}e=

3

13

⛦ Step-by-step explanation ⛦

\begin{gathered}\sf{\implies4x^2-9y^2=2ax+b^2}\\\sf{\implies4x^2-2ax-9y^2=b^2}\\\sf{\implies4\left(x^2+\dfrac{ax}{2}\right)-9y^2=b^2}\\\sf{\implies4\left(x^2-\dfrac{2ax}{4}+\dfrac{a^2}{16}-9y^2\right)=b^2-\dfrac{a^2}{4}}\\\sf{\implies4\left(x-\dfrac{a}{4}\right)^2-9y^2=b^2-\dfrac{a^2}{4}=k(say)}\\\sf{\implies\dfrac{\left(x-\dfrac{a}{4}\right)^2}{\dfrac{k}{4}}-\dfrac{y^2}{\dfrac{k}{9}}}\\\sf{\implies\:Let\:k > 0}\\\sf{\implies\:Therefore,\:e=\sqrt{1+\dfrac{\dfrac{k}{9}}{\dfrac{k}{4}}}}\\\sf{\implies\:e=\sqrt{1+\dfrac{4}{9}}}\\\\\boxed{\sf{e=\dfrac{\sqrt{13}}{3}}}\end{gathered}

⟹4x

2

−9y

2

=2ax+b

2

⟹4x

2

−2ax−9y

2

=b

2

⟹4(x

2

+

2

ax

)−9y

2

=b

2

⟹4(x

2

−

4

2ax

+

16

a

2

−9y

2

)=b

2

−

4

a

2

⟹4(x−

4

a

)

2

−9y

2

=b

2

−

4

a

2

=k(say)

⟹

4

k

(x−

4

a

)

2

−

9

k

y

2

⟹Letk>0

⟹Therefore,e=

1+

4

k

9

k

⟹e=

1+

9

4

e=

3

13