Find out the differential equation of the following two families of curves :
Straight lines having slope and $x$-intercept equal in magnitude.
Straight lines at a fixed distance $p$ from the origin.
My Approach :
- a straight line is defined by $y = mx + c$, $x$-intercept $= -c/m$ but $-c/m = m$, so $c = -m^2$. $$y = mx - m^2\; , \; dy/dx = m\; , \; y' = m$$
- Straight lines at a fixed distance $p$ from the origin : $$x\cos A + y\sin A = p,$$ $$y\sin A = p - x \cos A,$$ $$y = p\sin A - x\cot A,$$ $$y' = -\cot A.$$
Are my answers correct?