If the solution of differential equation \(\displaystyle\frac{dy}{dx} = \frac{ax + 3}{2y + 5}\) represents a circle, then a is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −2
The variables separate:
\( \displaystyle \int(2y + 5)\,dy = \int(ax + 3)\,dx \)
\( \displaystyle y^2 + 5y = \frac{a}{2}x^2 + 3x + C \)
\( \displaystyle -\frac{a}{2}x^2 + y^2 - 3x + 5y - C = 0 \)
A second-degree equation with no xy term is a circle when the coefficients of \(x^2\) and \(y^2\) are equal:
\( \displaystyle -\frac{a}{2} = 1 \)
\(a = -2\)