The curves for which the length of the normal is equal to the length of the radius vector is/are

Explanation

The length of normal ` |y| sqrt(y+((dy)/(dx))^(2))`
From the hypothesis of the problem
`|y| sqrt(1+((dy)/(dx))^(2))` = radius vector `= r = sqrt(x^(2) + y^(2))`
`rArr y^(2){1+ ((dy)/(dx))^(2)} = r^(2) = x^(2) + y^(2)`
`rArr y^(2) + y^(2)((dy)/(dx))^(2) = x^(2) + y^(2) " " rArr y^(2)((dy)/(dx))^(2) = x^(2)`
`rArr +- y(dy)/(dx) = x " " rArr +- ydy = xdx`
`rArr +- y^(2) = x^(2) - k^(2) " " :. x^(2) +- y^(2) = k^(2)`
This represents a circle or equilateral hyperbola according as we take + or - sign.