Let dy/dx = (ax - by + a)/(bx + cy + a), a,b,c ∈ R, represents a circle with center (α, β). Then, α + 2β is equal to :

Solution

Maths sir · 4 steps

Cross-multiply, spot x dy + y dx = d(xy), integrate, then force the result to look like a circle.

1. Cross-multiply and group. Write it as (bx + cy + a) dy = (ax - by + a) dx. Collect the b terms together: b (x dy + y dx) + cy dy + a dy - ax dx - a dx = 0, and note that x dy + y dx = d(xy).

   bxy + c/2y² + ay - a/2x² - ax = K

2. Conditions for a circle. A circle has no xy term, so b = 0. Its x² and y² coefficients must be equal, so c/2 = -a/2, giving c = -a, with a ≠ 0.
3. Find the centre. With b = 0 and c = -a, the equation is -a/2(x² + y²) - ax + ay = K. Multiply by -2/a to get x² + y² + 2x - 2y = K'. Comparing with x² + y² + 2gx + 2fy + c = 0 gives g = 1, f = -1, so the centre (-g, -f) is (-1, 1).
4. So. α = -1 and β = 1, so α + 2β = -1 + 2 = 1. The answer is (c).

The idea to keep. When the dx and dy terms mix x and y, look for exact pieces like x dy + y dx = d(xy). For a circle, kill the xy term and equate the x² and y² coefficients.

Answer from the NTA final answer key. Solution by PrepWiser.