Question

Find the differential equation of all circles passing through the origin and having their centers on the y axis

Answer 1

Equation of circle whose centre is (h, k)

(x – h)² + (y – k)² = r²

Since the centre is on the y-axis (ie) (0, k) be the centre

(x – 0)² + (y – k)² = r²

x² + (y – k)² = r²  ........(1)

Since the circle passing the origin (0, 0)

Equation (1) becomes

0 + (0 – k)² = r²

k² = r²

⇒ r = k

Equation (1)

⇒ x² + (y – k)² = k²

x² + y² – 2yk + k² = k²

x² + y² – 2yk = 0

x² + y² = 2yk  .......(2)

Differentiating w.r.t. x

`2x +  2y ("d"y)/("d"x) = 2"k" ("d"y)/("d"x)`

⇒ `x + y ("d"y)/("d"x) = "k" ("d"y)/("d"x)`

k = `(x + y ("d"y)/("d"x))/((("d"y)/("d"x))`  .........(3)

From (2) and (3)

`x^2 + y^2 = 2y ((x + y ("d"y)/("d"x))/((("d"y)/("d"x))))`

`(x^2 + y^2) ("d"y)/("d"x) = 2y(x y ("d"y)/("d"x))`

`x^2 ("d"y)/("d"x) + y^2 ("d"y)/("d"x) = 2xy + 2y^2 ("d"y)/("d"x)`

`x^2 ("d"y)/("d"x) - 2xy = 2y^2 ("d"y)/("d"x) - y^2 ("d"y)/("d"x)`

⇒ `y^2 ("d"y)/("d"x) = x^2 ("d"y)/("d"x) - 2xy`

÷ Each term by `(("d"y)/("d"x))`

⇒ `y^2 = x^2 - 2xy(("d"x)/("d"y))`