If the equation of the hyperbola with foci (4,2) and (8,2) is 3x2−y2−ax+by+γ=0, then a+b+γ is equal to:
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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 a2(x−h)2−b2(y−k)2=1, where (h,k) is the center.
Step 3: Relate a2,b2,c2 and eccentricity
For a hyperbola, the relationship between a, b, and c is c2=a2+b2. The eccentricity e is given by e=ac. We are given that the eccentricity e=2. Using this, we can find the values of a2 and b2.
Step 4: Substitute values into the standard equation
Substitute the center (h,k)=(6,2) and the calculated values a2=4 and b2=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 3x2−y2−ax+by+γ=0. By comparing the coefficients, we can find the values of a, b, and γ.
Step 6: Calculate a+b+γ
Finally, substitute the determined values of a, b, and γ into the expression a+b+γ to find the required sum.