Differential Equations Question 203
Question: The normal to a curve at P(x, y) meets the x-axis at G. If the distance of G from the origin is twice the abscissa of P, then the curve is a
Options:
A) parabola
B) circle
C) hyperbola
D) ellipse
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Slope of tangent $ =\frac{dy}{dx} $
$ \therefore $ Slope of normal $ =-\frac{dx}{dy} $ Thus, the equation of normal is $ Y-y=-\frac{dx}{dy}(X-y) $ This meets x-axis (y=0), where $ -y=-\frac{dx}{dy}(X-x) $ or $ X=x+y\frac{dy}{dx} $
$ \therefore G $ is $ ( x+y\frac{dy}{dx},0 ) $
$ \therefore OG=2x $
$ \therefore x+y\frac{dy}{dx}=2x $ Or $ y\frac{dy}{dx}=x $ or $ ydy=xdx $ Integrating, we get $ \frac{y^{2}}{2}=\frac{x^{2}}{2}+\frac{C}{2} $ Or $ y^{2}-x^{2}=c, $ which is a hyperbola.
BETA
