# Hyperbola equation

Find the hyperbola equation with the center of S [0; 0], passing through the points:
A [5; 3] B [8; -10]

Result

f = (Correct answer is: f = 7 * x^2 - 3y^2 = 148)

#### Solution:

$\dfrac{ (x-x_{0})^2 }{ a^2 } - \dfrac{ (y-y_{0})^2 }{ b^2 }=1 \ \\ \ \\ \dfrac{ x^2 }{ a^2 } - \dfrac{ y^2 }{ b^2 }=1 \ \\ \ \\ 5^2/a^2 - 3^2/b^2=1 \ \\ 8^2/a^2 - (-10)^2/b^2=1 \ \\ \ \\ 25/a^2=1 +9/b^2 \ \\ a^2=25 / (1 +9/b^2) \ \\ \ \\ 64/25 \cdot \ (1 + 9/b^2) - 100/b^2=1 \ \\ \ \\ \ \\ 64/25 \cdot \ (b^2 + 9) - 100=b^2 \ \\ 1.56b^2 -76.96=0 \ \\ \ \\ p=1.56; q=0; r=-76.96 \ \\ D=q^2 - 4pr=0^2 - 4\cdot 1.56 \cdot (-76.96)=480.2304 \ \\ D>0 \ \\ \ \\ b_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ \pm \sqrt{ 480.23 } }{ 3.12 } \ \\ b_{1,2}=\pm 7.0237691685685 \ \\ b_{1}=7.0237691685685 \ \\ b_{2}=-7.0237691685685 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 1.56 (b -7.0237691685685) (b +7.0237691685685)=0 \ \\ \ \\ b=b_{1}=7.0238 \doteq 7.0238 \ \\ \ \\ \ \\ a=\sqrt{ 25/(1+9/b^2) }=\sqrt{ 25/(1+9/7.0238^2) } \doteq 4.5981 \ \\ \ \\ x^2/(148/7)- y^2/(148/3)=1 \ \\ \ \\ f=7 \cdot \ x^2 - 3y^2=148$

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