If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3x^{2}-y^{2}-\alpha x+\beta y+\gamma=0$, then $\alpha+\beta+\gamma$ is equal to ____.

JEE Main · 2025 · 03 Apr 2025 · Shift 2 · Hyperbola

If the equation of the hyperbola with foci

(4, 2)

and

(8, 2)

3 x^{2} - y^{2} - α x + β y + γ = 0

, then

α + β + γ

is equal to ____.

Answer: 141

The foci

(4, 2)

and

(8, 2)

give centre

(6, 2)

and

c = 2

, with a horizontal transverse axis:

\frac{( x - 6 ) ^{2}}{a ^{2}} - \frac{( y - 2 ) ^{2}}{b ^{2}} = 1

and

a^{2} + b^{2} = 4

. Clearing denominators, the coefficients of

x^{2}

and

y^{2}

are in the ratio

b^{2} : - a^{2}

, which must match

3 : - 1

, so

b^{2} = 3 a^{2}

; with

a^{2} + b^{2} = 4

this gives

a^{2} = 1, b^{2} = 3

. The equation becomes

3 (x - 6)^{2} - (y - 2)^{2} = 3

, i.e.

3 x^{2} - y^{2} - 36 x + 4 y + 101 = 0

. Hence

α = 36, β = 4, γ = 101

, and

α + β + γ = 141

JEE Main · 2025 · 03 Apr 2025 · Shift 2 · Integer answer