Question

A curve passing through the point (l, l) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of point P from x axes. Determine the equation of the curve

Answer 1

hmmm, well the curve if it is perpendicular, then its angle must be 90. so hmmm, , well the origin must be equal to the 2 by tenth time of the curve. but whether it is perpendicular or not, it is not equal. so, the curve must be a undefinite proportion, which cannot be defined .

Answer 2

the equation of the normal passing through the origin
$x+ \frac{dy}{dx}y=0$
and
$y= \frac{|x+ \frac{dy}{dx}y|}{ \sqrt{1+ \frac{dy}{dx}^2 } }$
solve it
$y^2=x^2+2xy \frac{dy}{dx}$
$\frac{dy}{dx}= \frac{y^2-x^2}{2xy}$
solve it
take y=vx
u get
$x^{2} + y^{2}=cx$
it passes through (1,1)
[tex] x^{2} +y^2=2x[/tex]