Question

Find the equation of a curve passing through the point (0, -2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

Answer 1

Answer:

y² - x² = -4

Step-by-step explanation:

Hi,

Let any point on the curve be P(x, y).

Given that the product of the slope of its tangent and

y coordinate of the point is equal to the x coordinate

of the point.

Slope of tangent to any curve f(x) is given by

$\dfrac{dy}{dx}$ ,

So, given that $y*\dfrac{dy}{dx} = x$

$\dfrac{dy}{dx} = x/y$

We can rewrite above equation as

y dy = x dx

Integrating on both sides, we get

$\int y dy = \int x dx$,

$\dfrac{y^{2}}{2} = \dfrac{x^{2}}{2} + c,$

y² - x² = 2c = k(say),

Also given that curve passes through (0, -2),

Hence, 0 - (-2)² = k = -4

Hence, required equation of the curve is

y² - x² = -4 which is the equation of rectangular

hyperbola.

Hope, it helps !