Differential Equations Question 263
Question: 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
Options:
A) $ xy=1 $
B) $ xy=2 $
C) $ xy=3 $
D) None of these
Show Answer
Answer:
Correct Answer: B
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=\frac{dy}{dx}(X-x) $
It intersects the axis of x where Y=0 i.e $ -y=\frac{dy}{dx}(X-x) $ or $ X=x-y\frac{dy}{dx}=OT $
Hence $ x-y\frac{dy}{dx}=2x $ or $ \frac{dx}{x}+\frac{dy}{y}=0 $ Integrating, $ \log x+\log y=\log C $ i.e., xy=C. This passes through (1, 2)
$ \therefore C=2. $ Hence the required curve is xy = 2
BETA
