# 17.    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.

H Harsh Kankaria

According to the question,

$y\frac{dy}{dx} =x$

$\\ \implies \int ydy =\int xdx \\ \implies \frac{y^2}{2} = \frac{x^2}{2} + c$

Now, Since the curve passes through (0,-2).

x =0 and y = -2

$\\ \implies \frac{(-2)^2}{2} = \frac{0^2}{2} + c \\ \implies c = 2$

Putting the value of c, we get

$\\ \frac{y^2}{2} = \frac{x^2}{2} + 2 \\ \implies y^2 = x^2 + 4$

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