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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


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 .
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The equation of the normal passing through the origin
x+ \frac{dy}{dx}y=0
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)
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