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in Differential equations by (36.3k points)

A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is y2 – 2xy(dy/dx) – x= 0 and hence find the equation of the curve.

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Best answer

The tangent at P(x, y) is

Y – y = (dy/dx)(X – x)

If p be the length of perpendicular from the origin, then

Hence, the result.

Also, 

Integrating, we get

log |v2 + 1| + log |x| = log c

(v2 + 1)x = c

((y/x)2 + 1)x = c

x2 + y2 = cx

which is the required equation of the curve.

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