The curve for which the length of the normal is equal to the length of the radius vector is/are (a) circles (b) rectangular hyperbola (c) ellipses (d) straight lines

Options

A. circles
B. rectangular hyperbola
C. ellipses
D. straight lines

Answer

A, B

Explanation

We have length of the normal = radius vector
or `ysqrt(1+((dy)/(dx))^(2)) = sqrt((x^(2)+y^(2))`
or `y^(2)(1+((dy)/(dx))^(2)) = x^(2)+y^(2)`
or `x=+-y(dy)/(dx)`
i.e., `x=y(dy)/(dx)` or `x=-y(dy)/(dx)`
i.e., `x=y(dy)/(dx)` or `x=-y(dy)/(dx)`
i.e., `xdx-ydy=0` or `xdx+ydy=0`
Clearly, `x^(2)-y^(2)=c_(1)` represents a rectangular hyperbola and `x^(2)+y^(2)=c_(2)` represents circles.