A curve is such that the portion of the x-axis cut off between the | Filo

Class 12

Math

Calculus

Differential Equations

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150

A curve is such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2). The equation of the curve is

1. xy=1
2. xy=2
3. xy=3
4. None of these
Solution: [b]  Let P(x, y) be any point on the curve, PM the perpendicular to x-axis PT the tangent at P meeting the axis of x at T. as given OT=2 OM=2x. equation of the tangent at P(x, y) is Y−y=dydx(X−x) It intersects the axis of x where Y=0 i.e −y=dydx(X−x) or X=x−ydydx=OT Hence x−ydydx=2x or dxx+dyy=0 Integrating, logx+logy=logC i.e., xy=C. This passes through (1, 2) ∴C=2. Hence the required curve is xy = 2
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