Question

Find the differential equation of the family of all circles, whose center lies on the x-axis and touches the y-axis at the origin.

• A. $2xy \frac{dy}{dx} = y^2 - x^2$
• B. $2xy \frac{dy}{dx} = x^2 - y^2$
• C. $x^2 + y^2 = 2xy \frac{dy}{dx}$
• D. $x^2 + y^2 = 2y \frac{dy}{dx}$

Hint:

When differentiating implicit equations, apply the chain rule and remember to differentiate each term with respect to $x$. For a family of circles, center and radius conditions are key.

Answer 1

The Correct Option is A

The equation of a circle with center at $(h, 0)$ and radius $h$ is given by: $$(x - h)^2 + y^2 = h^2.$$ $$x^2 + y^2 + h^2 - 2hx = h^2$$ $$x^2 + y^2 - 2hx = 0$$ $$2x + 2y \frac{dy}{dx} - 2h = 0$$ $$h = x + y \frac{dy}{dx}$$ $$x^2 + y^2 - 2x \left( x + y \frac{dy}{dx} \right) = 0$$ $$x^2 + y^2 - 2x^2 - 2xy \frac{dy}{dx} = 0$$ $$y^2 - x^2 - 2xy \frac{dy}{dx} = 0$$Then, equation of the family of circles is: $$2xy \frac{dy}{dx} = y^2 - x^2.$$