Question

Form the differential equation of circles passing through the points of intersection of unit circle with centre at the origin and the line bisecting the first quadrant.

Answer 1

Given,

Equation of circle:-

$x^2+y^2=1$

line bisecting in first quadrant:-

$y=x$

$\therefore (x^2+y^2-1)+\lambda (y-x)=0$

$2x+2y{y}^{\prime }+\lambda (y^{\prime }-1)=0$

$\lambda =-\frac{2x+2y{y}^{\prime }}{y^{\prime }-1}$

$(x^2+y^2-1)-2\frac{(x+y{y}^{\prime })}{y^{\prime }-1}(y-x)=0$

$(y^{\prime }-1)(x^2+y^2-1)-2(x+y{y}^{\prime })(y-x)=0$

Is the required differential equation of circle.