JEE PYQ: Differential Equation Question 44
Question 44 - 2019 (10 Jan Shift 2)
The curve amongst the family of curves, $(x^2 - y^2),dx + 2xy,dy = 0$, which passes through $(1, 1)$, is:
(1) a circle with centre on the $x$-axis
(2) an ellipse with major axis along the $y$-axis
(3) a circle with centre on the $y$-axis
(4) a hyperbola with transverse axis along the $x$-axis
Show Answer
Answer: (1)
Solution
$(y^2 - x^2),dx = 2xy,dy$. So $\frac{d(y^2)}{dx}\cdot x - y^2 = -x^2$. Let $v = \frac{y^2}{x}$: $d\left(\frac{y^2}{x}\right) = -dx$. Integrating: $\frac{y^2}{x} = -x + C$. So $y^2 = -x^2 + Cx$, i.e. $(x-\frac{C}{2})^2 + y^2 = \frac{C^2}{4}$. Through $(1,1)$: $C = 2$. Equation: $(x-1)^2 + y^2 = 1$. Circle with centre $(1,0)$ on the $x$-axis.
BETA
