If the equation of the hyperbola with foci (4, 2) and (8, 2) is , then is equal to __________. [2025]

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Solution

Q.
If the equation of the hyperbola with foci (4, 2) and (8, 2) is $3 x^{2} - y^{2} - α x + β y + γ = 0$ , then $α + β + γ$ is equal to __________. [2025]

Ans.
(141)

Equation of hyperbola is $\frac{{(x - 6)}^{2}}{a^{2}} - \frac{{(y - 2)}^{2}}{4 - a^{2}} = 1$

$\Rightarrow (4 - a^{2}) {(x - 6)}^{2} - a^{2} {(y - 2)}^{2} = a^{2} (4 - a^{2})$

$\Rightarrow (4 - a^{2}) x^{2} - a^{2} y^{2} + (12 a^{2} - 48) x + 4 a^{2} y + 144 - 44 a^{2} + a^{4} = 0$

On comparing with $3 x^{2} - y^{2} - α x + β y + γ = 0$ , we get $a^{2} = 1$ and $α = 36$ , $β = 4$ and $γ = 101$

$∴ α + β + γ = 141$

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