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.

Solution

Let y = f(x) be equation of curve
Now

dy over dx

is slope of tangent to the curve at point (x, y).
From the given condition,

dy over dx cross times straight y space equals space straight x space space space space space space space space space space space space space space space space space space space space or space space space space space dy over dx space equals space straight x over straight y

From the given condition,

dy over dx cross times straight y space equals straight x space space space space space space or space space space dy over dx space equals space straight x over straight y

Separating the variables and integrating,

integral space straight y space dy space equals space integral straight x space dx space space space space space space space space space space space or space space space space space straight y squared over 2 space equals space straight x squared over 2 plus straight c

...(1)
Since it passes through (0, -2)

therefore space space space space space fraction numerator left parenthesis negative 2 right parenthesis squared over denominator 2 end fraction space equals space fraction numerator left parenthesis 0 right parenthesis squared over denominator 2 end fraction plus straight c space space space space rightwards double arrow space space space space 2 space equals space straight c

therefore space space space from space left parenthesis 1 right parenthesis comma space space space space straight y squared over 2 space equals space straight x squared over 2 plus 2 space space space or space space space space straight y squared space equals space straight x squared plus 4

which is required equation of curve.

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