Question

A curve passes through the point (0, -2) and at any point (x, y) of 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. Find the equation of the curve.

Answer 1

Given that the product of slope of tangent and y coordinate equals the x-coordinate i.e.,y \(\frac{dy}{dx}\) = x

We have ydy = xdx

⇒ ∫ydy = ∫xdx

⇒ y²/2 = x²/2 + c

For the curve passes through (0, -2), we get c = 2, 

Thus, the required particular solution is:-

∴ y² = x² + 4