Show that , the equation of all circles touchung the y -axis at the origin is `2xy dy/dx= y^2-x^2`.

Show that , the equation of all circles touching the y-axis at the origin is , 2xy (dy)/(dx) = y^(2) - x^(2)

Form the differential equation of all circle touching the x-axis at the origin and centre on the y-axis.

show that the equation of the parabola whose vertex and focus are on the x -axis at distances a and a' from the origin respectively is y^(2) = 4 (a'-a) (x - a )

Equation of the circle concentric to the circle x^2+y^2+4x-2y=20 and through the origin is

A circle of radius r passes through the origin and intersects the x-axis and y-axis at P and Q respectively. Show that the equation to the locus of the foot of the perpendicular drawn form the origin upon the line segment bar (PQ) is (x^(2)+y^(2))^(3)=4r^(2)x^(2)y^(2) .

Solve xy(dy)/(dx)=x^2+2y^2

Show that the general solution of the differential equation (dy)/(dx) + (y^(2) + y + 1)/(x^(2) + x + 1) = 0 is given by ( x + y + 1) = A(1 - x - y - 2xy) , where A is parameter.

The equation of the locus of the middle point of a chord of the circle x^2+y^2=2(x+y) such that the pair of lines joining the origin to the point of intersection of the chord and the circle are equally inclined to the x-axis is (a) x+y=2 (b) x-y=2 (c) 2x-y=1 (d) none of these

The differential equation of all the ellipses centred at the origin and have axes as the co-ordinate axes is where y'=(dy)/(dx), y''=(d^(2)y)/(dx^(2))

If a is a prameter , show that the differential equation of the family of curves y = (a-x)/(ax+1) " is " (x^(2) +1) (dy)/(dx) + y^(2) + 1= 0 .