Won't get to this part in class, the teacher can't help and it's due soon..? Please help if you can!

Question

A curve passing through the point(1,1) 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 P from the x-axis.Determine the equation of the curve.

Answer 1

At point P(x,y)

perpendicular from origin is such that

sqrt[y^2 + x^2] = y

y^2 + x^2 = y^2   => x=0.  

So, the perpendicular distance is y [at x=0] and the tangent at (0. y) is y=y` parallel to x axis {slope, m =0}.

Assuming

y = x^n + a1 x^n-1 + .......+ an &

y` = an.

Also it includes point (1, 1)

1 =  a1 + a2 + ......+y`

=>a1=a2=a/n-1/=0, y`=1

So, the curve is the line

y=1.