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What will be the differential equation of the family of circles having their centres on the y axis |
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Answer» Answer: The system of CIRCLES touching Y axis at origin will have centres on X axis. Let (a,0) be the CENTRE of a circle. Then the radius of the circle should be a units, since the circle should touch Y axis at origin. Equation of a circle with centre at (a,0) and radius a (x─a)²+(y─0)²=a² That is, x²+y²─2ax=0 ─────► (1) The above equation represents the family of circles touching Y axis at origin. Here 'a' is an arbitrary CONSTANT. In order to find the differential equation of system of circles touching Y axis at origin, eliminate the the arbitrary constant from equation(1) Differentiating equation(1) with RESPECT to x, 2x+2ydy/dx─2a=0 or 2a=2(x+ydy/dx) Replacing '2a' of equation(1) with the above expression, you get x²+y²─2(x+ydy/dx)(x)=0 That is, ─x²+y²─2xydy/dx=0 or x²─y²+2xydy/dx=0 |
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