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For the hyperbola 16*x² - 25*y² - 96*x + 100*y - 356 = 0, a tangent line making an angle of pi/4 with the transverse axis has the form y = x + lambda (with lambda > 0). Find the value of 2*lambda.

1. 2
2. 4
3. 6
4. 8

Correct answer: 4

Solution

Completing the square: 16(x-3)² - 25(y-2)² = 400, so (x-3)²/25 - (y-2)²/16 = 1. Center (3,2), a²=25, b²=16. For tangent y=x+c to standard shifted hyperbola: (c-2+3)²... use c' = a²*m - b²: tangent y-2 = 1*(x-3)+c' with c' = ±sqrt(25-16) = ±3. So y = x - 3 + 2 + 3 = x+2 or y = x - 3 + 2 - 3 = x - 4. For lambda > 0, lambda = 2, and 2*lambda = 4.

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