Differential Equations (Families of Curves): Circles with fixed radius r and tangent to the y-axis.

widdonod1t

widdonod1t

Answered

2021-12-27

Differential Equations (Families of Curves): Circles with fixed radius r and tangent to the y-axis.

Answer & Explanation

poleglit3

poleglit3

Expert

2021-12-28Added 32 answers

The general equation of a circle with centre at (h, k) and radius r is given by
(xh)2+(yk)2=r2
Also, it is given that y axis is tangent to this circle.
This means the perpendicular distance from the centre of the circle to the line x=0 is equal to the radius of the circle.
This means h=r
Hence, the equation of such a circle becomes
(xr)2+(yk)2=r2
Here, k is hte only variable left.
Hence, to obtain the differential equation, we need to eliminate the variable k
We will do this with the help of differentiation.
Differentiating the curve with respect to x, we get
2(xr)+2(yk)dydx=0
(yk)dydx=rx
(yk)=rxdydx
Putting the value of yk back in the equation we get
(xr)2+(yk)2=r2
(xr)2+(rxdydx)=r2
(dydx)2(xr)2+(xr)2=(dydx)2(r2)
(dydx)2(x2+r22xrr2)+(xr)2=0
(dydx)2(x22xr)+(xr)2=0
(dydx)2=(xr)22xrx2
Marcus Herman

Marcus Herman

Expert

2021-12-29Added 41 answers

(xa)2+y2=r2
2(xa)+2yy1=0
(xa)=yy1
y2y12+y2=r2

Vasquez

Vasquez

Expert

2022-01-09Added 457 answers

Center is (0,k) and radius is r.
Equation is
x2+(yk)2=r2 Equation 1
By differentiating we get,
2x+2(yk)dydx=0
k=+xdxdy+y Equation 2
Putting the value of K from Equation 1 and 2, we get
x2+(yxdxdyy)2=r2x2+x2(dxdy)2=r21+(dxdy)2=r2x2(dydx)2=x2r2x2

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