For a hyperbola, the relationship between a, b, and c is c² = a² + b². The eccentricity e is given by e = (c)/(a). We are given that the eccentricity e = 2. Using this, we can find the values of a² and b².

For a hyperbola, the relationship between a, b, and c is c² = a² + b². The eccentricity e is given by e = (c)/(a). We are given that the eccentricity e = 2. Using this, we can find the values of a² and b².

Substitute the center (h,k) = (6,2) and the calculated values a² = 4 and b² = 12 into the standard equation of the hyperbola.

Finally, substitute the determined values of a, b, and into the expression a+b+ to find the required sum.

If the equation of the hyperbola with foci $(4{,}2)$ and $(8{,}2)$ is $3x^2 - y^2 - ax + by + \gamma = 0,$ then $a + b + \gamma$ is equal to:

Answer: 132

Step-by-step solution

Step 1: Find the center and distance between foci

The center of the hyperbola is the midpoint of the segment connecting the two foci. Given the foci are $(4{,}2)$ and $(8{,}2)$ , we can calculate the center. The distance between the foci is $2c$ , which can be found using the distance formula.

Step 2: Determine the standard form of the hyperbola

Since the $y$ -coordinates of the foci are the same, the transverse axis is parallel to the $x$ -axis. This means the standard form of the hyperbola is $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ , where $(h,k)$ is the center.

Step 3: Relate $a^2, b^2, c^2$ and eccentricity

For a hyperbola, the relationship between $a$ , $b$ , and $c$ is $c^2 = a^2 + b^2$ . The eccentricity $e$ is given by $e = \frac{c}{a}$ . We are given that the eccentricity $e = 2$ . Using this, we can find the values of $a^2$ and $b^2$ .

Step 4: Substitute values into the standard equation

Substitute the center $(h,k) = (6{,}2)$ and the calculated values $a^2 = 4$ and $b^2 = 12$ into the standard equation of the hyperbola.

Step 5: Convert to general form and compare coefficients

Multiply the standard equation by the least common multiple of the denominators (12) to clear fractions, then expand and rearrange it into the general form $3x^2 - y^2 - ax + by + \gamma = 0$ . By comparing the coefficients, we can find the values of $a$ , $b$ , and $\gamma$ .

Step 6: Calculate $a+b+\gamma$

Finally, substitute the determined values of $a$ , $b$ , and $\gamma$ into the expression $a+b+\gamma$ to find the required sum.

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If the equation of the hyperbola with foci (4,2)(4{,}2)(4,2) and (8,2)(8{,}2)(8,2) is 3x2−y2−ax+by+γ=0,3x^2 - y^2 - ax + by + \gamma = 0,3x2−y2−ax+by+γ=0, then a+b+γa + b + \gammaa+b+γ is equal to: