`S(x ,y)=0` represents a circle. The equation `S(x ,2)=0` gives two identical solutions: `x=1` . The equation `S(1,y)=0` given two solutions: `y=0,2.` Find the equation of the circle.

The solution of the equation (x^2y+x^2)dx+y^2(x-1)dy=0 is given by

Find the value of k, if x = 2, y =1 is a solution of the equation 2x + 3y = k. Find two more solutions of the resultant equation.

Find the parametric form of the equation of the circle x^2+y^2+p x+p y=0.

A person standing at a junction (crossing) of two straight paths represented by the equations 2x-3y+4=0 and 3x+4y-5=0 seek to reach the path whose equation is 6x-7y+8=0 in the least times. Find the equation of the path that he should follow.

Find the equation of hyperbola having foci S(2, 1) and S'(10, 1) and a straingt line x+y-9=0 as its tangent. Also, find the equation of its director circle.

Find the particular solution of the differential equation (dy)/(dx)=-4xy^(2), given that y=-1, where x=0

Find the particular solution of the differential equation (dy)/(dx) = - 4xy^(2) given that y = 1, when x = 0.

Find k if the equations x+2y+2z=0, x-3y-3z=0, 2x+y+kz=0 have only the trivial solution.

Let y_(1) and y_(2) be two different solutions of the equation (dy)/(dx)+P(x).y=Q(x) . Then alphay_(1)+betay_(2) will be solution of the given equation if alpha + beta=……………….