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

Name: Differential Equations (Families of Curves): Circles with fixed radius r and tangent to the y-axis.
Uploaded: 2022-02-23T17:22:57+00:00
Description: Differential Equations (Families of Curves): Circles with fixed radius r and tangent to the y-axis.

Question

poleglit3 · Accepted Answer

The general equation of a circle with centre at (h, k) and radius r is given by  (x−h)2+(y−k)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  (x−r)2+(y−k)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(x−r)+2(y−k)dydx=0  (y−k)dydx=r−x  (y−k)=r−xdydx  Putting the value of y−k back in the equation we get  (x−r)2+(y−k)2=r2  (x−r)2+(r−xdydx)=r2  (dydx)2(x−r)2+(x−r)2=(dydx)2(r2)  (dydx)2(x2+r2−2xr−r2)+(x−r)2=0  (dydx)2(x2−2xr)+(x−r)2=0  (dydx)2=(x−r)22xr−x2

Marcus Herman · Accepted Answer

⇒(x−a)2+y2=r2 ⇒2(x−a)+2yy1=0 ⇒(x−a)=−yy1 ⇒y2y12+y2=r2

Vasquez · Accepted Answer

Center is (0,k) and radius is r. Equation is x2+(y−k)2=r2 Equation 1 By differentiating we get, 2x+2(y−k)dydx=0 k=+xdxdy+y Equation 2 Putting the value of K from Equation 1 and 2, we get x2+(y−xdxdy−y)2=r2x2+x2(dxdy)2=r21+(dxdy)2=r2x2(dydx)2=x2r2−x2

star233 · Accepted Answer

Answer:x=a±r2−y2Explanation:To solve the differential equations for circles with a fixed radius r and tangent to the y-axis, we can start by considering the equation of a circle with center (a, b) and radius r, which is given by:(x−a)2+(y−b)2=r2Since the circle is tangent to the y-axis, it means that the center lies on the x-axis, which implies b = 0. Substituting this into the equation above, we get:(x−a)2+y2=r2To find the family of curves that satisfy this equation, we need to solve for x in terms of y. Let&#039;s proceed with the solution:(x−a)2+y2=r2Expanding the square term, we have:x2−2ax+a2+y2=r2Rearranging the terms, we get:x2−2ax+(a2+y2−r2)=0This is a quadratic equation in x, and we can solve it using the quadratic formula:x=−(−2a)±(−2a)2−4(a2+y2−r2)2Simplifying further, we have:x=2a±4a2−4a2−4y2+4r22x=a±r2−y2Therefore, the equation of the family of circles with radius r and tangent to the y-axis is:x=a±r2−y2

alenahelenash · Accepted Answer

Step 1:To solve the differential equation for circles with a fixed radius, r, and tangent to the y-axis, we can start by considering the equation of a circle with center (a, r) and radius r. The equation of such a circle can be written as:(x−a)2+(y−r)2=r2Since the circle is tangent to the y-axis, the x-coordinate of its center, a, will be equal to the radius r. Therefore, we have:(x−r)2+(y−r)2=r2To find the family of curves satisfying this equation, we need to eliminate the square root. We can do this by squaring both sides of the equation:(x−r)2+(y−r)2=r2Expanding the equation, we get:x2−2xr+r2+y2−2yr+r2=r2Simplifying the equation, we have:x2−2xr+y2−2yr=0Step 2:Now, we can factor out r from the terms involving x and y:r(x−2)+y(x−2)=0Factoring out the common factor of (x - 2), we obtain:(x−2)(r+y)=0Setting each factor equal to zero, we have two possible cases:Case 1: x - 2 = 0In this case, we find that x = 2. Substituting this value back into the original equation, we get:(2−r)2+(y−r)2=r2Simplifying the equation further, we obtain:4−4r+r2+y2−2yr+r2=r2y2−2yr+r2=0This equation represents a single point, (r, r), which is the point where the circle is tangent to the y-axis.Case 2: r + y = 0In this case, we find that y = -r. Substituting this value back into the original equation, we get:(x−r)2+(−r−r)2=r2Simplifying the equation further, we obtain:(x−r)2+4r2=r2x2−2xr+r2+4r2=r2x2−2xr+4r2=0This equation represents a parabola with its vertex at (r, 0) and axis parallel to the x-axis.In conclusion, the family of curves representing circles with a fixed radius r and tangent to the y-axis consists of a single point (r, r) and a parabola with its vertex at (r, 0) and axis parallel to the x-axis.

Differential Equations (Families of Curves): Circles with fixed radius 'r'

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