isi-entrance 2020 Q6

isi-entrance · India · UGB Implicit equations and differentiation Eliminate parameter from implicit family and derive ODE

Prove that the family of curves
$$\frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} = 1$$
satisfies
$$\frac{dy}{dx}\left(a^{2} - b^{2}\right) = \left(x + y\frac{dy}{dx}\right)\left(x\frac{dy}{dx} - y\right).$$

Prove that the family of curves

$$\frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} = 1$$

satisfies

$$\frac{dy}{dx}\left(a^{2} - b^{2}\right) = \left(x + y\frac{dy}{dx}\right)\left(x\frac{dy}{dx} - y\right).$$

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8