Question

Form a differential equation of the family of circles having a centre on y-axis and radius 3 units.

Answer 1

The equation of the circle passing through the point (O,k)and radius 3 is of the form

(x - 0)² + (y - k)² = 9

⇒ x² + (y + k)² = 9 .........(i)

Differentiatitng w.r.t to x we get;

2x + 2(y - k) y₁ = 0 ⇒ x + (y - k)y₁ = 0

⇒ x + (y - k)y₁ = \(\frac{x}{y_1}\)

Substituting in (1) We get ;

⇒ x² + (\(-\frac{x}{y_1}\))² = 9 ⇒ x² + \(\frac{x^2}{{y_1}^2}\) = 9

⇒ x²y₁² + x^{2 =} 9y₁²