Question

The normal lines to a given curve at each point pass through (2,0). The curve passes through (2,3). Formulate the differential equation and hence find out the equation of the curve.

Answer 1

Let P(x,y) be any point on the curve. The equation of the normal at P(x,y) to the given curve is

$Y-y=-\frac{1}{\frac{dy}{dx}}(X-x)$

Normal at each point pases through (2,0). Hence,

$0-y=-\frac{1}{\frac{dy}{dx}}(2-x)$

$y\frac{dy}{dx}=2-x$

$ydy=(2-x)dx$

$\frac{y^2}{2}=-\frac{(2-x{)}^2}{2}+C$

$y^2=-(2-x{)}^2+2C$

This passes through (2,3). Therefore,

$9=0+2C$

$C=\frac{9}{2}$

Therefore, the equation of the required curve is $y^2=-(2-x{)}^2+9$